I would say that the probation period is going to vary, mainly because of two
very important factors. How old the joining cat is, and what lifestyle they are coming from.
Kits probably wouldn't have any sort of proper probation period. The Warrior Code forbids placing kits in harms way, so for the most part it would be considered unthinkable to
not allow an orphaned kit to join and immediately assign a foster mother to look after them in the nursery. Although the tricky thing with kits joining is making certain that they're actually orphans and not just lost or even runaways. Even unintentionally stealing children is a great way to piss off your neighbors. So if a kit is found and brought back to the camp, I would think most leaders would have enough commonsense to gently question (Or get the Medicine Cat to do so when they check over the kit) the kit about where they came from and why they're on their own. The deputy would also probably want to absolutely sure that every base has been covered, so the odd patrol or two would be sent out scout out the kits story and to spread the word about said kit being found on so-and-so Clan's border.
But if the kits story is demeaned plausible, and the patrols come back to camp without anyone claiming to know the kit, then the kit would be renamed (If applicable) and handed over to a trustworthy Queen to foster. (If the kit is too young to speak, then obviously no questioning would occur, but in this case, extra emphasis would presumably be placed on the Deputy to send out patrols to spread the word of a kit being found.)
Apprentice age cats are kind of hard to judge, but I would assume they also wouldn't have a long probation period. I would say putting it at about half a moon would be fair in most cases. What makes apprentice age cats so hard to judge is that they're at an in-between stage of development. They aren't kits anymore, but they aren't adults yet either. So care needs to be taken with regards to noting their decision making process. They certainly will need time to make such a drastic decision, but on the other hand, considering they'll be going thru a full apprenticeship once they do join, their shouldn't be much of a need to determine if they're trustworthy.
Really I think the half moon would be more about allowing the cat to make friends amoungst their peer group and to determine they're learning style so the most appropriate mentor can be assigned to them. Certainly these cats would be questioned about their origins and watched over carefully during this short time, but they still aren't adults, so the Clan overall isn't going to judge them too harshly. I can definitely this age group being taken out by various different adult cats to get a feel on who they respond best to, and to determine what they known skill-wise. So perhaps completing a series of tasks before joining proper and being re-named would be on the table, but taking part wouldn't determine if the cat can actually join, just what will be best for them if they do join.
Now as for adult cats, they're definitely going to be treated more like cut and dry cases. I would say a moon would be a fair probation period for these cats. Theirs no need to fuss over their origins like kits, and as adults, they'll generally be expected to know the basics or hunting and fighting at the very least. Really what this probation period will focus on is determining if this cat is trustworthy and if their capable of finding their place in the Clan. Because until Kits and Apprentice age cats, these adult cats will have their ways and personality set pretty firmly in stone, so they'll need a much longer adjustment period then a younger cat would. Plus the Clans generally hold the view that their society is much more "civilized" or at least better then what other groups of feral cats have, so an outsider cat that's already an adult when they want to join is going to be judged a great deal more harshly and have their motives for joining scrutinized.
But the general idea is still the same- Hear the cats story, let them have chances to prove themselves or at least be useful, allow them to make friends and get absorbed in Clan life overall during this probation period. Adult cats will just be watched
much more closely for signs of untrustworthyness. But if all goes well, then by the end of the moon the cat would be offered the chance to became a member of the Clan permanently and be renamed.
So to recap- Kits: A day, if not even that. Apprentice age: Half a moon. Adults: A moon. Although sometimes I can see these ages not being cut and dry. A cat that's five months old is still considered a "kit" by Clan law, but this "kit" may very well act more like an apprentice since life outside the Clans probably doesn't allow for a lengthy childhood. A cat that's 9 moons old is still considered an "apprentice" by Clan law, but this "apprentice" could very well be past what is typical for a Clan apprentice of the same age to know skill-wise, for much the same reasons I gave for the "kit". With adults it's much less murky, but these kind of gray areas would certainly need to be considered by the Clan Leader when they assign probation periods to incoming cats.
Now we can get into lifestyle! Another minefield. But depending on how one headcanons Loners, Rouges, Barn Cat and Kittypets... Their probably won't be as much thought given to previous lifestyle as their is to age, but it should still be taken into account.
I headcanon Loners as being well... loners. These cats are most likely to be nomadic, by themselves (Or in a pair or trio at the most), and not terribly inclined to claim and protect a territory (And playing more realistically, they're also more likely to be Toms.) As such, these cats would probably have some difficulty settling into a Clan and the Clan would be less likely to view them as trustworthy given their previous independence. Loners would probably join up as either young apprentice age cats or as older adults.
I headcanon Rouges much differently then what Canon depicts them as. These cats are most likely to have a set territory, be a family unit, and very eager to claim, expand and protect a territory (Also playing more realistically, I mostly see related Mollies and their dependent kits as Rouges.) Since these cats already live in a group based society, they would probably have a much easier time settling into a Clan, but depending on how many Rouges join at the same time, the Clan might be weary of them retaining stronger bonds to one another rather then bonding to the Clan overall. Rouges would probably join at all ages.
I practically defined what Barn Cats are on as a whole because Canon can't decide what they are. So they are a little different overall from other groups of outsider cats. I see them as having a territory (A Rouge trait), living in a family unit (Another Rouge trait, but I personally believe Barn Cats live in much smaller groups. And they may even have a Tom or two living with them),
not being overly fussed about expanding and defending said territory (A Loner trait) and tolerating humans (A definitely a Kittypet trait. In fact, as the name implies, these cats are most likely to live in or close to a human structure.) Much like Rogues, since these cats lived in a group before, they'll likely have an easier time settling into a Clan. But also like Rogues, the Clan might be worried about them bonding to the Clan overall. Another thing that might cause issues is the Barn Cats tolerance (Or even worship) of humans. The Warrior Code prohibits Clan Cat-Human interaction whenever possible, so the Barn Cats might have some difficulty in learning to accept humans as potential enemies now, and the Clan isn't likely to be thrilled about them viewing humans semi-decently in the first place. Barn Cats would probably join at any age, but it would likely take something happening to their living situation to get them to abandon their previous home and join a Clan.
Kittypets don't require much headcanoning, although I personally believe that the Clan cats would be mystified by their kit-like ways well into adulthood. (Since domestic animals are in fact well known to retain juvenile behavior even after sexual maturity) So depending on the current human culture regarding the keeping of pet cats- Kittypets either have never even been outside, or have a smallish "territory" that includes their owners garden and possibly a little ways away from it. These cats are by far the most likely to live with non-relatives, or even by themselves. Despite Canon annoyingly building up Kittypets as masters of territory defense, with their rather sedate lifestyle, regular meals and lack of practice, I can't see them as being able to hold their own against the average Clan cat. Kittypets will by far have the hardest time adjusting to life in a Clan unless they were taken in as a kit or a relatively young apprentice age cat. In fact, given their lifestyle being so "unnatural" in the Clan cats eyes, adult Kittypets may not even be allowed to join the Clan. Unless said Clan has a Daylight Warrior program running (Or is willing to start one up) already. Kittypets could join at any age, but preference will definitely be given to young cats.
So this is a heck of a lot to consider, so any kind of cat- no matter how old they are or where they came from- is going to cause quite a stir by joining a Clan. It's definitely not something you can apply a one size fits all approach to, all of my recommendations are generalizations at best. Their the baseline and the Leader should go from there. But generally yes, kittypets aside, a moon should be a long enough of a probation period for adult cats.
Also thank you! My first thought with Millie was to name her
Silver something, but given Silverstream... I would understand Graystripe not being comfortable with that choice and Firestar knowing that fact. Minnows are still lovely and silvery in color, so Minnow- Millie became. (Plus my person preference for completely renaming cats is to keep the letter of the original name.) -stripe has nothing to do with Graystripe, but rather because I believe Millie has been described as "thickly" or "unusually" striped, and that's what -stripe is for.
As for Purdy... I don't know, but Pheasant just came right to my mind when I thought of him. They are brown for one, and Purdy is such a lovable old coot that I associate him with birds because well... I've met many a sweet old lady or gentle old man feeding the birds in my local park. So the elderly = birds to me apparently. Plus pheasants are very pretty birds and Purdy has always struck me as a cat that was probably exceedingly handsome in his younger days. -pelt is just for him being a short-hair.
Also thank you! My first thought with Millie was to name her Silver something, but given Silverstream... I would understand Graystripe not being comfortable with that choice and Firestar knowing that fact. Minnows are still lovely and silvery in color, so Minnow- Millie became. (Plus my person preference for completely renaming cats is to keep the letter of the original name.) -stripe has nothing to do with Graystripe, but rather because I believe Millie has been described as "thickly" or "unusually" striped, and that's what -stripe is for.