I've worked at the opposite side of this, as the stage manager on a bar / small venue. It's a bit different from coffee houses, but some of the information might be useful for you. The bar I worked usually had small, local bands, we had a lot of first timers, but we also had famous (as in "internationally famous") bands play at our stage to pay us back for giving them support at the beginning of their careers.
1) How do you get gigs?
By sending in a demo to the email address on our web page, by calling booking (a hard feat as they do their best to keep their numbers hidden from prospective bands) and try to get a gig or by word of mouth (someone working at the venue saw you somewhere else and loved your style / make up / lead guitarist and told booking).
Pestering anyone else at the venue, or even pestering booking usually hurts more than it helps unless you're completely new (in which case we'll blame ignorance, not stupidity or desperation, and usually set you straight). Now if you apply to every opportunity we have for a band of your genre, that's just a good thing. At our venue we have several two-day festivals for unsigned bands of different genres (punk, pop, metal) as well as two-three Thursdays a month for Rock N Roll Thursdays (for all new / unsigned / inexperienced small bands who wants to play on a stage). Now this might not be how most others handle it, but it was how we did it.
2) Do you have to pay to play a gig? / What do you earn playing a gig?
Depends on which country you're in. In England (especially in London) it's common to pay a fee to be given stage time. In most other countries (Norway and the US, for example) this is unheard of. One American artist I asked was almost offended by my question, asking why he should pay to do his job. Now what you earn is really dependent on how famous you are (and sometimes how good you are / how much the ones booking you like your music), but I have never heard of anyone having to pay here. That's money going the wrong way.
On the other hand, this doesn't mean a band starting out is being paid much. And by "much" I mean "anything at all". I know my venue used to give them 24 beers to share and nothing else (this is considered being generous to small bands, so other venues can give them even less). We've sometimes paid part of their travelling costs, so they wouldn't go broke performing, but I've never heard of a band earning anything the first years of their career. In fact in Norway you don't do it for the love of money, and even nationally famous bands like Enslaved have second jobs to keep things afloat.
I know that sometimes when booking falls in love with a foreign band, they will go together with venues elsewhere in Norway to split the bill for getting them to Norway and around the country. The band will not be paid anything besides transportation fees, though.
3) How are requests handled?
Badly. Very badly.
I'm kidding, but only somewhat. Bands don't like being told what to play. No, not even if its their own song and you really, really, really love it. Really small bands might still be flattered that you know the title of one of your songs and play it as a reward, but most others will curse you out then and there, or wait until they get backstage. I've been at concerts where requests were made and responded to (most notably Roger Waters and Travis), but mostly the band ignores the request makers. Roger Waters told the young boy he had excellent taste, but they unfortunately hadn't played it in so long he didn't want to risk it, and the singer of Travis started laughing, wondered loudly if they still remembered that song and played it because it was one of the first songs they made and they were flattered anyone remembered. Every single other concert I've been at where requests were made, they were either met with insults or just completely ignored.
If the band has asked for the audience to give requests, that usually don't fare much better. Unless the band is a cover band, playing other people's songs, they don't want to play Smoke on the Water or Stairway to Heaven. (Not to mention playing them on stage actually leads to you owing money to the original bands, which is a pain.) If the band asks for requests, most will either be "funny" ("play something by Smokie!") or just miss the point completely, asking for songs already played or something equally stupid.
This might just be me being an old grump, and being wrong, but rock / metal gigs rarely have requests, and rarely get requests actually worth playing.
4) Forcing the band off stage?
If it's an open mik night, forcing a band off stage with booing can be done. Some places even have a ritual where the bar's owner or some other authority after some booing makes a big deal out of chasing the band off stage. Where I work, that has never happened. We book bands for a certain amount of time, and we won't chase them off stage until they've finished their set.
Now if they go over their allotted time, that's a whole new deal. First the stage manager / sound guy will give them a sign that this is their last song (holding up a single finger). If they keep going past that song, we'll give them a sign to stop playing (drawing our fingers across our throat). If they keep going, the stage manager lets the sound guy know to cut the sound. This means the only sound heard out is the drums, drowning out the other music, and they look like asses. It's not a good way to stop a concert, and it's only done if there's many bands playing and a tight schedule.
These bands will be told off that playing more than their alloted time don't increase their fee, and might even have a harder time getting a new gig at that place because they can't obey such a simple rule. Going over their time despite signals is rude as hell, and it pisses of the next band and everyone working with them at the venue. How well-loved a band is does not influence their playing time, but if they follow the rules and the audience love them they'll probably get more stage time the next gig. Not to mention it almost guarantees them a second gig at that venue. Always leave them wanting more!
If they, on the other hand, are completely hated by the audience, that can lead to so much. Most of the time the audience just don't give them any feedback, but sometimes someone throws bottles (a practice the venue I worked at tried to stop by kicking out anyone who did this). I know that at Reading festival there's a sport in chasing unpopular bands off the stage by bottling them, but that's a rain of bottles, not just a beer glass or two. As a safety feature we don't give out glass bottles during gigs, but give them flimsy plastic glasses that can't hurt. The best advice I was given on my first concert was "Never bottle the band. More likely than not, they have better aim than you.", and I have seen artists throw back and artists climb down and let their fists do the speaking. (Of course they spoke directly to my face, giving me the best shiner I've ever had, as I was working as stage crew that evening and tried to break up the fight.)
The only times a concert has been stopped by the audience were times when they managed to get beer over the sound booth, short circuiting some of the equipment. Note that this is rarely taken well by the audience or the venue, and you will find yourself banned for life as well as given a hefty bill to replace what you broke.