What happens when a player that you set as an override in a specific formation isn't available (injury, fatigue, etc)? Does the regular starter play instead?

For example, in I Formation 3WR I override WR3 with my TE1, but my TE1 gets hurt. Does the regular WR3 come in for those plays?

Yes.

What if the #1 player is also fatigued or injured. Would the #3 player take over?

It will go down the depth chart until it finds someone. If it doesn't find anyone it will reduce the fatigue threshold and try again. If for some reason it never finds someone that way, it picks the best player based on your weights.