"didn't" is correct as the apostrophe is used where the letter is missed out - the apostrophe replaces "o" of "did not".
It is alright to say "ice-cream" bit "ice cream" without a hyphen is still better and more common in the UK.
It sounds a bit odd to say "I liked my birthday because there was a horse there." If someone said to you: "Where was the horse?" The reply "on my birthday" would seem really odd. You need to reword this.
It's nice to know that you are taking the trouble to ask. Well done. A bit puzzled by the question of the horse though! I agree totally with RevShirls and you could perhaps say that you enjoyed your "birthday PARTY because..." Or have I missed the point?