The King's Mirror mentions that an armored horseman may bring a bow or crossbow into battle, if they want. Could a knight really choose to bring a bow instead of a lance? Would an archer-knight still be deployed with the rest of the heavy cavalry or in a separate role?

by hborrgg

The relevant passage:

there are, however, other weapons which a mounted warrior may use, if he wishes; among these are the "horn bow" and the weaker crossbow, which a man can easily draw even when on horseback, and certain other weapons, too, if he should want them.

http://www.mediumaevum.com/75years/mirror/sec2.html#XXXVII

Also, how common was it for a knight to be as well equipped as described in the King's Mirror? ie armored horse, a mail shirt worn over a breastplate plus two gambesons, two swords, etc.?

Muleo

Could a knight really choose to bring a bow instead of a lance?

That's not how I read it at all. It says he must have a lance (spear) and then goes on about other optional weapons.