In fights, mostly it's just experience which improves. This is more of an invisible stat, but it's what determines the trainers advice (e.g need to fight brawlers, boxer punchers or whatever). The other stat which improves with fights is Stamina, which goes up by completing rounds.

Sparring is another way to increase experience, and happens by default in fight camp. If you don't select anyone yourself, the game chooses fighters around same weight class in the same gym to spar with.  If you want to spar outside of fight camp, you have to set this up yourself.

Hope the above makes sense...

Also, do boxers spar with amateur boxers as well or only with pros? Because I have one boxer whose gym don't have any other pro boxers to spar with.

If you don't have anyone to spar with and haven't manually invited anyone, it just uses basic default sparring partners which don't do as much good.

In training, fighters keep improving until they either reach their max potential for their stats (pre determined and invisible to you) or they go "on the slide". There isn't a fixed age for going on the slide, it depends on weight class, training stat, age and how much damage they've taken in their career. E.g heavier weight classes with good training stats might fight into their 40s, but lighter weight classes rarely make it past mid 30s

Once on the slide, certain stats start to diminish and there is an increasing chance every month that they might decide to announce their retirement