Did I use my light fixture's ground wire correctly?
It sounds like this is what you're describing...



If this is indeed what you've encountered, you should remove the grounding conductor from the twist-on wire connector used to connect the white wires. 



The only place grounded (neutral) conductors and grounding conductors should be bonded (connected), is at the main service disconnect.

If there are no grounding conductors in the house, it's not that big of a concern. As long as the wiring is done properly, you should have no problems at all.  If there truly are no grounding conductors in the house, connecting fixture grounding conductors will have no effect. If you're really concerned about the lack of grounding conductors, you can install combination ground-fault circuit interrupter (GFCI) breakers in the service panel.



This sentence also has me concerned...

I cannot use the metal ceiling box that was being used with the ceiling fan. I can only use the bracket, which is currently screwed into place, right into the "hole" in the ceiling.

Are you saying that the new light fixture is not installed in an electrical box?

If this is the case, you should correct that issue as well.  All electrical connections must be made inside an approved, listed and labeled enclosure.