Weather can be pretty unpredictable, last couple of winters it actually snowed quite a bit in Rome. Just be prepared. How to dress.....only thing I can say is, whatever you like. Not really a certain dress code and italian girls dress pretty wild sometimes. Since it will be winter when you go, there's even less to worry, especially for seeing churches, the Vatican and such. You'll be covered up
What else to see there I couldn't tell you, I lived near Pisa, made it to the embassy in Rome, that's all.
Riding should be possible, definitely. Ask for "Centro equestre" or "Maneggio". I hope you speak a little Italian. Not very many Italians speak English. Or at least they won't speak it out of fear to being wrong.
I will see if I have anybody on my Facebook group from Rome who could help you at least with the riding. Posted via Mobile Device