Hello,
There are a lot of questions, that's ok.
1 - I'd like to know what I would need in a Cisco UCS world to replace this functionality, namely, can I shift all L3 routing from the 3750 stack to the FIs?
No, the FIs run only as L2 switch, you will need a L3 device to perform routing for your enviroment.
2 - Can I connect, at leas temporarily, the HP DL380 servers directly to the FIs?
Yes, if you have the correct transceiver you can define the port as appliance mode to do that. (remember each FI is a different switch, they don't talk each other directly).
3 - Can I connect the FIs directly to my edge switches or ASA5520 cluster?
Yes, if your firewall will perform all L3 function of your UCS enviroment.
4 - Perhaps I should keep the 3750X stack for 1GB connections, and get 10GB modules to provide 2-4 10GB uplinks between the FIs and the 3750X stack?
This is a good idea.
Think the UCS not just like a common server deployment, but like a complete solution to manage your servers, L2 network and SAN. If you look deep will see a lot of good things that can help you in your daily operation. (Service profiles, callhome, firmware management, system visibility, etc, etc....).