I've run my own club in SL for several years; like nearly all clubs in SL it is free to enter but we rely on donations (tips) to keep going. Most DJs and clubs use tipjars - not necessarily a jar, sometimes it's a board with the pic of the DJ or host on it, sometimes it's a quirky little object, it will usually have a bit of hovertext over it to say how much has been donated so far that session, and sometimes how much the last donation was and who paid it.
There will usually be two or three tipjars; one for the DJ or singer, one for the venue itself and one for the host, if there is one. (The host is the person who greets everyone, offers to get them a group invite and makes other announcements pertaining to the event or the club. It's also the host's role to keep conversation going during the event and, if necessary, deal with troublemakers). The tipjars for the DJ and Host will usually be located close to where they are standing. If you cant find them, a quick IM to the host to ask will be fine - that's exactly the kind of task they are getting paid for.
The amount of tips vary widely, from just a few L$ and upwards, though Sylvia is right; the average for a DJ is about L$100, and the average for hosts and for the venue itself is about half that (meaning generally that half as many people will tip the host as well as the DJ, but those who do, will tip about the same amount). Some people like to tip in smaller amounts at intervals, say L$25 every half-hour; that way their tip reflects the length of time they stay in the club (and indirectly, how much they are enjoying it).
Tipping isn't mandatory but it is definitely appreciated; as a club owner I get no other income to keep the club running and any shortfall in land fees and other expenses has to come from my own RL pockets. Likewise the DJ has their own expenses; most DJs rent a stream to broadcast with, and they also need to pay for legal downloads of the music they play, as well as for the time spent in putting together and presenting their playlist, finding request songs etc. As a club owner I would not be upset if someone new comes and does not tip. It's more important to me that they enjoy the music and the conversation, and come back again, maybe bring a friend next time. Perhaps next time you'll be able to afford to give a little, even if not this time.
Regarding the music not playing while you were at the club, sometimes you need to "toggle" (turn off and on again) your music player to pick up the new stream. If you can't hear it in-world, go to the World > About Land menu in the Sound tab, you can copy the stream url and paste it into your regular media player or even in your external browser. Still dancing after you leave the club - that's because to stop dancing you need to click the dance machine again - sometimes there is a menu popup and you have to choose "Stop" and sometimes it just stops automatically. If you forget, and you're still dancing when you get home, try Me > Movement > Stop animating me from the menu at the top. That doesn't always work though; if it doesn't, you'll need to re-log.