With regard to "any tune", the Common Metre of ECD is a 32-bar jig or reel arranged as 4 "lines" of 8 bars each. Other (less) common formats are 40 and 48 bars (5x8 & 6x8). Many dances - especially American Contras - aren't fussy in the least what music they are danced to, as long as it fits. Other dances do seem prefer particular tunes, especially if they were originally composed for them. Some dance/tune associations, though, are nothing more than long-established custom.

I think it's a good idea to stick to these associations, partly because of "conditioned recall": it's easier to remember things in the circumstances under which you learned them (which is why a lot of Morris sides practise at pubs, or so they claim). Remembering a dance is thus helped by hearing the same music every time. Incidentally I was once in a display side which made a point of occasionally practising dances to "wrong" tunes, in order to disable dancers' autopilots so they had to think. Newcastle to a Steeleye Span instrumental, anyone?