It is incorrect to say that the loop-gain must be stable in order for the complete, closed loop system, to be stable.
Note: in the system \$ F = \frac{G}{1+GH} \$, loop gain is \$GH\$, not \$1+GH\$, as you stated.
It may be true that instability in the path \$GH\$ could cause closed-loop instability (for example, if \$H\$ itself is an ill-behaved entity), but as you noted, \$GH\$ appears in the denominator of the closed loop transfer function, and any notion of stability pertaining to \$GH\$ in and of itself is rendered moot, once you close the loop.
Rather than aiming for stability in \$GH\$, you are more concerned with obtaining a magnitude and phase response of \$GH\$ with frequency that does not cause feedback to become simultaneousy positive with a gain of one or more at any frequency. In response to your question about determining the shape of \$GH\$, that is the main purpose of Bode plots, graphs of gain and phase vs. frequency, from which you can visually identify such conditions.
I think it's fair to say (although I am not certain that it is formally correct to say this), that those required conditions for phase and magitude of \$GH\$ (to obtain stability) correspond to the requirement that the complex roots of \$1+GH\$ all reside in the left half of the complex plane (they all have negative real parts).
The fact that the denominator of the closed loop transfer function is \$1+GH\$, as opposed to simply \$GH\$, means that the poles and zeroes of \$GH\$ do not become zeroes and poles respectively of the closed loop transfer function, although I imagine that this could be a reasonable approximation for certain instances of \$G\$ and \$H\$.