People request a few songs, say a really slow song, then a really fast song, jumping from rnb to dance? How do you beatmatch those

No real reason to - the D.J runs the Disco and not one person from the audience. From my own point of view I wouldn't let myself be put in a situation where I would go from a Dance or Trance track, for example into a slow R'n'B number since from a dancefloor point of view, it wouldn't exactly do me or them any favours .

Better to concentrate on a more subtle process, perhaps over a few tracks, rather than changing from one type of music to something completely different.

dito the above you should always try to blend you music as much as possible imo

Do your best to play as many requests as possible, but, as mentioned above, you're the only one who should decide "when" then get played.

If someone requests an R'n'B track (say 100bpm) and a dance track (say 130bpm) while you've just started a 60's selection, tell the requestor that you "should" be able to play both "sometime between now and the end of the night". If he/she throws the old "My taxi turns up in 10 minutes" line on you, apologise that, in that case, they might not hear them (both).

There are certain times where playing a "crasher" is exactly what you want mind you. If you've been told that the buffet, or raffle, or presentations, or speeches etc..etc..etc are to be held in 5 mins..by all means announce "Tracy's request for bohemian Rhapsody" it'll save you the Anti-DJ'like announcement of "clear the dancefloor please" , a few mins later.