Biting is very common and I wouldn't consider it 'fighting'
Axolotls are very impulsive eaters, movement can sometimes cause them to instinctively 'SNAP' at the 'food' be it a worm, pellet, tank buddy or finger, as you will no doubt discover at some point!
Some axolotls are snappier than others (I have one that has a 3ft tank all to himself because he's very bitey) but no experience of mine would ever suggest it is hunger or viciousness, just instinct. It happens.
Indian Almond/Catappa leaves are wonderful and I highly recommend them. They last a few weeks and eventually break up and make a profound mess, but they will soothe a wound and will create a slightly antibacterial effect.
As always, keep the tank cool and the water clean, but it sounds like your beautiful axolotl is doing his thing and healing nicely.