I'm not going to pretend to know the answer, but atm I barely understand the question. What, to you, defines a battle as being medieval? If you could elucidate what exactly you're looking for, you might get a better answer.

What is the last medieval battle fought?

You really really need to define that question better. Are you asking about the medieval period in which case whatever year you decide counts as the the end of the medieval period will have had battles take place in it.

If you are looking for battles that take place without firearms its going to turn into a question about the definition of battle and may hit issues with the 20 year rule.

If you are looking for battles that take place with medieval style tactics with a direct decent from medieval armies then the Battle of Flodden is a good a candidate as any. While both sides used canon the english were very much using their medieval style bill and bow tactics where as the the scotish were attempting to use renaissance style pikes.

What is the battle where firearms are widely-used for the first time and serve us a standard weapon?

Two different questions. Early hand canons may well have been a fairly standard weapon amoung those who could afford them while only being deployed in fairly small numbers. With europe early firearms evidence is limited are limited to a few pictures in manuscripts, a few entries in accounts and the odd archaeological find. They were clearly around but its hard to say how widely they were used (In the 2005 Journal of the Ordnance society is an article that talks about a lot of the evidence around early European guns).

If you are looking for shear numbers then the development of the Pike and shot formations in the the pike and shot formations around 1500 are the first large scale standardised units to deploy arquebuses in large numbers. That lead to the Tercio which was nominally 1/3rd arquebuses.

From there a range of factors including better guns and bayonets leads to a move to more guns and less pikes and swords. The Battle of Breitenfeld is a good a candidate as any for the beginning of the dominance of firearms.