First question: LEGALLY MEL is not required. However, I consult it inflight to assess the effects. With multiple failures you're earning your money by going beyond SOPs and NNC. Now you have to actually think
Q2: our Cat3 checklist (used before ToD) mentions "Aircraft Qualified". This implies to check MEL restrictions. I've compiled an easily accessible list from MEL which I keep handy.
Q3: same as Q1: consult, note the FL250 limitation and make your decision. If you climb to planned cruise level, be VERY cautious with anti ice now...
If the route involves long isolated areas,, think twice.
Consider this: your failure has no NNC implications, only MEL. Destination has no engineering. You can't contact OPS. What would you do?
Whether a company fires you depends more on the company then on your decisions, if you know what I mean.