1. You can adjust the numbers of champions in game creation options.

2. Some quests reward you with new champions.

3. If you want more champs - play as Kraxis. Their ability allows you to hire heroes of both kingdom and empire alignment. Or play the Altars - they can train henchmen that are actually adjustable champions.

There is a cheat that 'produces' an offspring of your sovereign that will mature in several turns. Note, however, that this is dynasty related cheat and dynasties are not present in FE. The cheat combination is something like ctrl + b. You`ll have to write 'cheat' in your game`s shortcut/steam launch options.

Yes, champions are not very common. It could be that I see only a small part of the map in the first few hours, so there are probably like 20 heroes on the map, even if I get to see only three.

One of the higher military techs grants me a unit that can level up like a champion (says the description) - I cannot build them, because I have zero metal. Once I get some I will see what they are worth.