Hi, welcome to the forum!
The timeframe is a month. Some authors have arranged contracts with CoG to have long-term private WIPs. Maybe some Patreon authors knowingly or unknowingly break that rule without signing a contract, but I’ve no idea.
No, the hosting of WIPs is totally separate.
Well, overall it’s for authors to have more financial support while they’re writing and for their fans to give them that support, either to feel good about it or to access some kind of extra reward (whether that’s an earlier version of the WIP before other people, extras, etc, depending on the Patreon owner’s decisions).
In practice, I don’t recommend keeping playtesting entirely/mostly paywalled because it reduces the pool of testers and so it’s easier for bugs to go unnoticed; it also selects for people who are already fans of the project so can result in overlooking other issues.
I also strongly encourage potential subscribers to do their research when deciding if they want to spend money on supporting an author (that isn’t simply buying their game when it’s out). Without signing up it’s easy to scroll through and see how frequently the Patreon owner posts, and the kinds of things they’re posting. If you’re hoping for access to an early version of a WIP and they’ve not posted one in a year, you may do well to spend your money elsewhere.
Also it’s helpful to keep in mind that most WIPs so not get completed. Someone having a Patreon means they’re making some more money, but unless it’s so outlier-level successful that it means they don’t have a day job, it has less impact on whether the game will be released than you might think - especially if they haven’t released anything before.