Can the Fermi energy lie into the band gap? Fermi energy $\rightarrow$ highest energy level filled at $T=0K$
Fermi level $\rightarrow$ Energy level where we have a chance of $50\%$ to find an electron.
Now in my course text they say that for an insulator, the Fermi energy is inside a band gap. But since this zone is forbidden, how can this be the Fermi energy, if this Fermi energy is the highest occupied zone, how can it lie inside a forbidden zone?
So I assume that it is the Fermi level that will lie inside the band gap? But then, if this is a forbidden zone, how can you have $50\%$ chance of finding an electron if this zone is forbidden?
My last question, if the Fermi level is inside the band gap, does the Fermi energy lie in the band below the band gap? 
 A: You can think about the Fermi energy ($E_f$) as the the energy required to add or subtract one particle from the system (the chemical potential). As you define it so, you won't have any problem. When the Fermi energy (or the chemical potential) lies inside the band gap, you just see the many body effects have moved the energy required to add or subtract a particle from the system has changed. 
A: Part of the confusion arises because (not necessarily different) authors use the words Fermi level and Fermi energy inconsistently. Some use them interchangable (usually in the first sense), some use them the way you defined it (it is also common to use the term chemical potential instead of Fermi level).
In your nomenclature:


*

*In a prototypical semi-conductor (one condution band and one valence band), at $T = 0$ the Fermi level lies exactly in the center of the bandgap (and slightly moves around when you heat your system). The Fermi level defined this way is the chemical potential of the electrons.

*The Fermi energy will be the highest energy in the valence band (as that is the highest energy level occupied at $T = 0$).
But don't let authors confuse you by their inconsistent use of the terms. In calculations the Fermi energy usually does not appear (it is rather a qualitative feature of Fermionic systems), the Fermi level does appear, as the chemical potential in the Fermi distribution.
On to the question how the probability of finding the electron can be 50% if there are no states there: This statement is only true if the density of states is constant. The actual probability of finding an electron at energy $E$ is the Fermi distribution (which has the value $0.5$ at the Fermi level) multiplied by the density of states. If you now turn on temperature, the number of particles has to be conserved, as the Fermi distribution is symmetric, the Fermi level has to be in the middle of the bandgap at $T = 0$ to assure this.
Properly formulated, in a semi-conductor the Fermi energy will lie on top of the valence band (this language is necessary, as saying the Fermi energy lies in the band usually means that the system in question is a conductor). There again is a problem with inconsistent language. When saying the Fermi energy lies in the band gap, what is meant is this: It lies at the highest possible energy of the band (or in my terms on top of the band).
