I also think being able to change weapons would be great. It is very annoying to run behind mounted archer, especially if you have other footmen in your team and you see them running faster than you but not taking into account that the horse archer will only stop when he is in corner.
Some ideas of my own, concerning arena fights:
If you get knocked out, computer should NOT just assume your party lost! If there is one archer left, you have two lancers with you and the archer scores a lucky head-shot, he would still have no chance!
Arena rewards should depend on the balance between the sides. Simplest way would be to give each equipment type different value depending on how powerful the combination is. This should also affect the amount of gold you win from bets. No longer shall you receive just the 10 and 38 denars for defeating those mounted archer and lancer with sword, shield and suicidious two-handed-sword-wielding team mate.
Horses should remain longer after their rider is knocked out. I once shot a rider and went for his horse, but when I got to it it was evaporating. It was a strange moment, and first time I tried to mount enemy horse.
Maybe you should get bonus reward if the fight was interesting for the audience. Fight would be interesting if it lasted long, both sides would be very weary (all down/1-2 barely standing) and there were no long, boring moments where no blood flows. The problem with this are the people who will get some fast money by shooting all their friends and then firing out one arrow at every few minutes to make the fight last as long as possible without it validating it as "boring". Another type of interesting battle might be one that is completed very fast. I think it would require one of the participants to have downed at least two enemies and surviving unscratched, while the enemy got owned. Both "epic" and "heroic" fights would give you bonus reward, but wouldn't affect bets. It could be like normal 10, epic 15, and heroic 20 denars - you would only get the heroic bonus if you were the hero.
Now, a little story.
I think this was a 3v3 match. I was a footed bowman. My companions got killed, but enemy still had 2 men standing. At least one of them was mounted, possibly both. I killed one of them when he rode past me, and quickly took his horse. One last enemy was a lancer without his horse; I'm not sure if I had killed his horse or if he had lost it earlier. And the tricky part? The lancer had a shield. It took me about 40 arrows until I destroyed his shield. It had about 8 arrows in it, but it fell to ground inside-up, so I couldn't count them. After that battle I took Horse Archery and was very proud of myself.