Do you expect a raid slot?
Because, like Sellys I've earned one, so yes. When slotted as a regular raider in a regular schedule raid, of course. If it's made known from the start that some people might be swapped out, then it might be a different issue.
Do you feel that sometimes people are going to haft to sit out?
It happens, but I don't personally believe it's necessary. Current content no longer favors 2 tanks, 1 rogue, 5 healers, and 17 ranged dps. Most everything can be completed completely with a composition that works on every fight with minimal changes. Duel spec is key for times when fewer tanks or healers are necessary.
How does sitting out make you feel?
Been there, but it's varied. It was different back in former content, but now reasons vary. If it's simply to rotate players so more get a chance, fine, cool. If it's for one boss because someone's been trying to get a specific drop, that works, so as long as I don't need anything as well. If it's because a different class/role would be needed, I'd prefer to have the prerogative of using an able alt, having my character lose the spot, not me as a player. I show up on time, as scheduled and accepted, I don't want to sit around on the chance I might be needed. Once I'm declined a spot, fine, I'll either pug or lead an alt run for the lockout period. (see rant in next answer)
How would you set up a raid if you had, lets say, too many dps and only enough tanks and healers to support one raid group?
Schedule another raid at a separate time if there are alts that can fill the roles. Recruit tanks. Give priority to players willing to swap roles and/or use alts to make another group happen. Hell, I even have a second 85 tank, wouldn't take much to gear them up and simply tank for two groups.
Even better than getting another 10 man going:
SWAP TO TWENTY FIVE MAN RAID.I actually had a falling out with a guild after it was made clear that the GM and RL (two people) are the only ones that would ever run raid groups. That by no means would someone be allowed to set up another group or an alt run, and implied that if I was to do a pug run, I better be joining a pug, not starting one. This was absurd. Members sitting on the sidelines shouldn't be denied the chance to raid. Nor should they only be able to resort to PuGs started by random strangers. If you're not strict on recruiting specifically for certain raid spots, you have no business telling your members that if they aren't allowed in your raid group, they're not allowed to raid. Godforbid at 1 a.m. on a Tuesday morning (Monday night) you don't have a sizable pool of unsaved raidbots to draw from.
How you would handle the fact that the people you run with one night will most likely not be the same people your run with the next?
By being an elitist jack-[donkey synonym] and expect no less from them than from their predecessors. Lack of experience is one thing, but everyone has just as much access to strats/guides/videos. Might take a bit to learn specifics for the group, but it shouldn't be reinventing the wheel sort of labor every single week. If it's a random 9 people each week, it's harder. If I'm able to get a few regulars it helps more to set an example of consistency.
Generally speaking, of the topic: I tank Firelands. This is the first tier of content I ever had my cherry popped as a tank. From Kara through BoT, I entered first as DPS, and through OS gearing and just having proper raid awareness at all times, switching to tank was rather easy.
I've usually been one of the top fury warriors on this server, going all the way back to TBC when non-tank warriors barely existed in raids. Switching to tanking being first and foremost still feels weird at times. But I do it because I can. Me tanking means another group can happen. Me tanking means having a raid spot is more likely. Me tanking means less wipes, and more dead bosses and more purples for everyone! If I were told I was being denied my spot because some other player transferred over or left another guild and has full BiS gear was going to be given it, I'd point out plainly that I didn't fail at my role, that those behind the raid invites were looking to be carried, and that that sort of people I don't want to raid with anyways. Incidentally, I already don't roll with those sort of people.
I raid with the mindset that everyone should know what they're doing. That anyone, after several weeks of the same content, should be knowledgeable enough to lead their role. That after a few months, they should be able to tell anyone new to the group exactly how a fight goes.