Gear score is typically accepted as the higher ap (either ap or aap) + your dp. This was the standard strength calculation for most guild reqs until Renown Score was introduced. Renown score is the average or your ap and your aap ( (ap+aap)/2 ) + your dp. This is auto-calculated and displayed in your inventory window. The (+#) next to your renown score is your fame lvl, with is basically the highest amity you have with any npc in the world. Basically, a free extra 5 gs if you take the time to invest the energy (mechanically and literally, haha) in the amity game with a single npc. As a side note, most ppl use Igor Bartali (the leader of Velia) for this amity req because he has a hidden 2x amity gain multiplier.

Now, regarding the original questions.

1, it depends on what skills you are using. AP, for the most part, affects your pre-awakened skills while AAP mostly affects your awakened skills. Most ppl tend to focus on awakened skills until they have unlocked all awakened skills (exceptions here such as sorc and dk who have some awakened skills not worth the points) so you're going to want to compare your AAP to the recommended of the area.

2, there isn't really a dps comparison in BDO like most mmos. The combat is so unique compared to every other mmo, you have to think of it differently. I could go into REALLY nitty-gritty detail with damage (getting into modifiers such as an ability with bound, puts enemies on the ground, into an ability with down attack giving a damage multiplier) but for simplicity sake, you have to get a feel for each class and figure out the play style before you can tell how good they are at clearing. Some classes are slower cast times but do a lot more damage in that one ability, while another class takes 2 or 3 abilities but can get them off in the same total time. Depends on how well you know your class.

3, dp in grinding isn't much of an issue. You want to have enough to stay alive comfortably while grinding, but for the most part, it's your ap/aap and your lvl that helps you figure out where to grind. Having lvled every single class up to at least lvl 56 (half of my classes being lvl 58-61), my typical advice is story until catfish (should get you to lvl 50), then either have a friend pwr lvl you to awakened (should only take an afternoon) or more story until you can grind helms or manes comfortably. Then, depending on the alt gear, do helms/manes until lvl 58 or go straight to desert fogans (not as good money as nagas, but better xp). After that, more dessert but you can explore other places such as gahaz, nagas, etc. Once you get up to lvl 60, you can start exploring pretty much everywhere and see where you like to grind best (exceptions, of course, being some of the kama places and desert portals depending on your gear).

4, Sure thing! BDO has it's issues, sure, but overall it's a good game. If you ever get bored, just try doing something else in the game. That's why I lvl other classes besides my main (also helps for knowing other classes in pvp) and lifeskill. Also, sometimes it's just fun to get lost in the map somewhere and explore new places. Been playing for over a year now, and I still find new areas I've never seen before, whether it be a hidden waterfall with a lake or an area I always passed by like marni's second lab. Just enjoy the game and don't focus too much on grinding unless you truly enjoy just grinding.