Understanding transformer ratings
Primary: 220V Secondary: 12+12V Rating: 5A


This notation generally means that there are two secondaries, each rated at 12V and 5A. It could also mean a center-tapped secondary that's rated at 24V @ 5A overall.


  If I use it as a 24V transformer, is the total current 2.5A?


No, if you put the two secondaries in series, you get 24V @ 5A.


  What happens (besides heating) if the transformer's current rating is
  exceeded?


Isn't heating bad enough? Transformers generally aren't all that good at getting rid of heat (high thermal resistance between the windings and ambient). Once the internal insulation breaks down (or catches fire), it will fail fairly catastrophically because of the shorted-turn effect.