A faulty thermostat can cause a lot of issues, rough running would not be one of them however. Replace the thermostat with a genuine Motorcraft r Mazda part, the aftermarket junk ones just don't work. They open at the wrong times, etc. I've been through enough on my own rides.
My guess though is that since you say you have to pump the gas at startup, is that you have a leaking injector. I recommend doing a fuel pressure leakdown test to check for leaks. The system should meet the correct pressure with the KOEO. With the engine off, the fuel pressure should hold at that same pressure for at least 30 minutes, ideally it should hold overnight. If it fails this test, the pressure is leaking somewhere, with the most likely culprits being the check valve in the pump, the fuel pressure regulator, or at least one of the injectors.
If you replace the injectors, get a set of flow matched rebuilt injectors. I got a set once on eBay, worked great at a great price, but make sure they are flow matched. While you are at it, replace all the intake gaskets and inspect all the vacuum lines.
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