Does healing a hurt ally increase ÜberCharge rate?

The charge rate is doubled. However, this bonus is automatically applied during setup time, even on fully overhealed teammates, and does not stack. Thus, teammates damaging themselves is not necessary during setup time.

For the Medigun:

The ÜberCharge meter will fill faster, at 2.5%/second, when healing injured teammates or teammates whose health is not at 150%, at 1.25%/second.

The Kritzkrieg builds charge 25% faster, so the rate for healing injured teammates is 3.125% and for healthy teammates is 1.5625%.

Note that this refers to ÜberCharge. All that mumbo jumbo about faster healing rates depending on how long someone has been injured, refers to the rate of HP healed rather than ÜberCharge rate.

Yes. Also, the longer an ally goes without taking damage, the faster you will heal them.