Could the stat or the cap be what is causing the bubbles, and overflow issues?
It could, but it's probably not.
Seems pretty obvious the previous owner dumped it knowing they had a big debacle on their hands and you bought it with existing headgasket issues.
But let's assume I'm wrong:
First - let's clarify "causing bubbles" might be:
The stat and cap are not causing bubbles to be *introduced into cooling system full of coolant*
The stat and cap could potentially cause overheating which causes "bubbles" by boiling.
We need to determine if the bubbles are being introduced or an artifact of overheating.
1. top off the coolant in the radiator and get all the air out
2. monitor coolant loss - check the radiator coolant level every time before you drive and see if you're loosing coolant.
The goal is to see if the radiator level is dropping without overheating.
Once it overheats then there's no telling how it lost coolant since it may have pushed out, boiled out.
If it's constantly streaming bubbles from a cold start and you can't burp it - then you have a leak or headgasket failure.
If you finally get it topped off and full with no more bubbles and the coolant level drops each day in the radiator - you're loosing coolant somehere. Find the leak, or it's consumign it through the headgaskets.