Exercise
Let K be a field and d be a non-constant polynomial in K[X]. Then the element g in K[X]/(d) is an inverse of f in K[X]/(d),
if fg = d in K[X].
if fg = 1 in K[X]/(d).
if gcd(g,d) = 1.
No, then the product is 0 in K[X]/(d).
Yes, that is the definition of an inverse element of f.
No. What does it mean that f is invertible?