All of the problems you're having are related to the fact that eggplant stores a lot of moisture.
If you intend to fry it, you have to drain or blot the moisture first or it will make much mischief. This manifests itself in a number of ways -- flying hot oil; soggy, greasy eggplant; and eggplant sticking to the pan among them. In any case, it's a truism that eggplant must be "cooked" twice in order to get rid of the moisture, which is quite bitter in most species. Sometimes "cooking" takes the form of salting and draining. It's important to remember, if you do salt and drain, not to rinse the salt off -- or the eggplant will absorb the rinse water. Then, if you want to fry it, it's a good idea to dust or dredge the eggplant in flour in order to prevent it from absorbing oil.
As to the grill pan method, I fully agree with Mezzaluna. You certainly can spoil eggplant by grilling it too hot. The appropriate heat is probably a well-preheated pan on a medium or perhaps medium-low flame. Grilling over high heat is usually a method to get a certain crust texture and a rare interior. Since you want neither, high heat is probably not a great idea.
When you put the eggplant on the grill, don't be in a hurry to turn it. The less cooked, the more likely it is to stick. I'm a bit surprised cupcake brought it up, it's usually a male problem. With most foods, if you allow them to brown properly they'll turn easily. Very wet foods, like eggplant, are somewhat more problematic -- which takes us back to starch dusting. The flour holds the water and helps prevent the eggplant from sticking.
Note that Dillbert actually does cook his grilled eggplant twice. First he cooks the exterior on the grill, than he finishes it by baking in the oven. A good method, indeed.
Restaurant cooks use high flame on higher output stoves, yes. But for specific purposes which are usually more related to time saving in preheating, rather than actual cooking.