There's no need to downgrade if you want to unlock and jailbreak the phone, you can just do it with pwnagetool/blackra1n and blacksn0w.
But to answer your question, yes, you should be able to downgrade to 3.1, but not 3.0 or 3.0.1
There's no advantage to downgrading to 3.1 though, at best you'll get nothing and at worst, you'll get some bugs.
I personally didn't have any bugs at 3.1 but as far as I know, there were some battery issues, and some users would have their phone randomly turn off.
Also a few more things that I can't remember right now ;p
I doubt going to 3.1 will fix anything. What I'd do, is make a custom firmware with PwnageTool and do a clean restore with it through iTunes and then set the phone up as new phone, not restore from backup.
Then install blacksn0w through Cydia if you need the unlock. That's the best advice I can give you.
Are there wifi and battery issues on a jailbroken phone? I would try to do a fresh restore to the Apple 3.1.2 firmware, re-jailbreak and re-unlock using blackra1n/blacksn0w and see if that helps. I would not restore from an old backup as this has caused problems for some.