1/ Cobra needs its own CFW to run too.
2/ After downgrading you're not forced to stay on E3 CFW, but if you install another CFW you MUST make sure it has lv1 hash checks patched/disabled or you WILL brick your console.
I'm not sure if TB and Cobra CFW have the lv1 hash checks patched, but you can always email cobra and request them to add that.
PS3 Slim CECH-3004A 160GB (500GB)
PS2 Slim SCPH-70004 - FMCB 1.8b