Having participated in a number, it all depends on how you wanna do it.

Artists who've done it solo, I've seen it set up as "I'll do x amount of hours, and keep going as long as people keep donating for more art"

Some artists have collaborated so it's more of a communal livestream where folks drop in and out as time goes on, that the hosts vet and invite.

Some folks do it over a series of days, so they'll do "x amount of slots a day for y amount of days", with the usual caveat of as long as there's enough folks interested, the stream resumes the next day.

It all depends on your preferences.

What's the common protocol for getting money to the charity? Is it all collected by the artist who then publicly donates it, or do the donors donate to the charity directly and send proof of donation? How does one check to make sure no one's falsifying the donations?

At least in my experience, they use Tiltify, which lets the host know where it's donated to. I'm not too familiar with the inner workings on that side.
However, I've also seen where the artist collects the money and then donates it to the charity. It can be a whole tricky thing.

Oh great, Tiltify looks like something to look into, thank you!

The other solution seems tricky, yeah, but I imagine with enough transparency, good faith, and doing things live on stream, it can be made trustworthy.