(a) is correct.
For (b), you need to invoke more than the factorization. Namely, is not the splitting field because there are complex roots of the polynomial and this field is real. The only other possible proper subfield is , but if it is indeed a proper subfield of , then it does not contain and so cannot split the polynomial. Thus is the splitting field.
For (c), everything but your calculation of is correct. You seem to have added instead of multiplied. . But since the minimal polynomial for epsilon is still of degree 4 and similarly. Thus .
You multiply the degrees because the basis is basically the product of bases, i.e. .