Anything can happen, but here's what's fairly typical at venues that call themselves folk clubs (as opposed to sessions, which are less formal)

Do people gather on a stage to play or sit around in a circle or random folks play at tables?
Usually there's a small stage or an end of the room designated as such. And seating accordingly, with or without tables.

Is there generally a theme
Not usually, but some clubs like to do that, especially when it's a singer's night, perhaps if there's a seasonal excuse.

Can you just show up and play?
Yes, though if it's a club that has guests and you want to do that on a guest night and you aren't known to the organisers it's not a bad plan to phone up in advance.

Is it generally a weekly thing or nightly
Weekly, bi-weekly or monthly

Most folk clubs have booked guests, rarely every time if it's a weekly club. So they are a bit more organised that you seem to think, and have a schedule with dates etc. They also have an MC who runs the evening and they charge (not a lot) for entry.

But there are exceptions to almost all of that...
- "folk clubs" that call themselves that but don't have guests
- clubs where people tend to sing from wherever they're sitting
- concert clubs that don't have floor singers
- clubs with no charge but a "voluntary" collection
- "clubs" that put on concerts at random infrequent intervals

And then there's the things that we at least in the UK) don't usually call clubs, like sessions (song, instrumental or both), acts simply booked to perform at pubs and other venues, Arts centre, theatre and village hall concerts, "open mic" nights etc.

We don't have house concerts very much, because most of us don't have houses big enough. And similarly we don't have the coffee house culture that the USians seem to.

Anahata