I agree, you have a mess.
The fact that it is from a collection agency and this late may be a clue that there were several attempts to collect (to some now long past address), and it was turned over to someone who would financially benefit if they collect.
I guess I would first want to know if the company contacting you is a collection agency from the Country you were resident in at the time of the travel, or possibly the administrator used by Florence for the tickets. This late in the game, it should be the former, rather than the latter.
Second, yes, verify the infractions are valid.
Beyond that, whether to pay or not, I can't advise, but you might work to settle with the collection agency, indicating that if they cannot provide proof, you can not in good faith pay. Settling, based on proof, for a fraction of the stated amount is not unheard of.
As for time limits of notification. From what I recall, the limits apply differently depending if you are an Italian Resident, an EU resident, or Non-EU. If you are a Non-EU resident, you have fewer protections, and sadly, fewer avenues for recourse, complicated by a language barrier. On the flip side, the Penalties for non-compliance are not as severe as if you are an EU resident.