Yoyobarn, your counterexample for F_1 does not work since 0/2=0/1=0 so it is contained in F_1. You need to look at what the multiplicative inverse looks like in this case. Is it of the same form? You cannot think of these fractions as ordered pairs since some of them are equivalent.
To figure out why F_2 doesn't have a multiplicative inverse and why F_4 has a multiplicative inverse just continue your calculation:
What if this is equal to 1? Express c and d in terms of a and b, this way you get a general formula for the inverse. Obviously you wouldn't have time to do this on the exam but it's worth doing since it gives more insight than just picking counterexamples.
Most of the times expressions involving integers do not have integer inverses. But there are exceptions, for example the group of integer matrices with determinant 1.