1) For me it was in stages; originally it was basic survival, then it was surviving more and more difficult embarks (at the time in 40d surviving on a glacier was very hard and on a tundra/desert was almost impossible, its a lot easier now). These days its usually combat via some mods (I'll note them below) and creating harder and harder 'rules' to follow (no danger rooms of course, few or no traps, no walling in, etc).
2) Like you my focus is on fortress mode, never got into adventure mode, perhaps once its gets fleshed out some more. As for 'best packs', that is very much a matter of opinion. I never got into genesis, didn't like having different types of dwarves (particularly the fire guy who sometimes burns things). My favorite pack I guess would be dig deeper (gold) mostly cause I was in love with dig deeper back when, it adds some additional difficulty to the game, but there are a lot and I haven't tried most of them. As for tilesets, you can have whatever tileset you want with whatever mod pack, I prefer ASCII these days (I use Cheepicus, which is ascii but with a different dwarf image, I mainly use it because the colors are altered a bit and are easier on my eyes and its 'squared') but when I was newer I used Ironhand and Mayday, but again its just a matter of opinion.
As for general mods, I use fortress defense II which radically increases the sieges in terms of how early and how often them come as well as adds in many races, most of which are about the same or weaker then goblins (though often more numerous) but some later ones are extremely hard to kill. I also use as mentioned dig deeper and sometimes play with total conversions like Kobald Kamp and whatever the HP lovecraft one is called.
3) The most important thing to me is challenge. I like to make a difficult situation more difficult through rules and limitations. A lot of that relates to military. When I was new I was like a lot of people in that I relied on traps, walling myself in, and similar tactics to give me a huge advantage over the enemy. I don't do this anymore, while my interior fortress can have differences my surface is usually a courtyard with a wall surrounding it and a three-wide opening with floodgates as my only security (many creatures can destroy floodgates). I never (or at least very rarely) use traps and rely completely on my military to defend me. Also I only use the "lock my floodgate and hide!" option as a last resort, normally I just use it to give my military a moment to get positioned then open her up and let the body parts sail. As for military tactics.. Depends a lot on what you are fighting but generally speaking I go with at least two squads of melee to every one squad of ranged (10 per squad ultimately, fewer at the start obviously), having them training every other month. When an attack comes I can put my ranged up on the wall and let them shoot down into the courtyard (and also hit the enemy as they advance on me) and have my melee meet the enemy and attack them directly while the ranged rain bolts down upon them.
4) I don't use pre-built sets. While my embark can change depending on the situation or what challenges I've imposed, in general I have one doctor (2 points in each medical skill), one broker (starts with points in various skills related to bartering). In the beginning these two will cover broker, doctor, manager, bookkeeper and odds are leader. They can divide down once migrants arrive. In addition I usually have dwarves with various military skills and two 'teaching' points (I don't know if it actually does anything but I like to think it does) and one dwarf who is a decent armor/weaponsmith (5 pts in each).
As for supplies.. That also varies but generally I take some picks, axes, food and booze of course, a few materials of each 'mood' type (I hate losing dwarves to suicide) like sand or glass, leather, gem, etc, just enough to cover me until I can find or buy some more later. I also take some animals (usually goats or sheep) in case of food emergency (a few animals to slaughter can mean the difference between abandoning and surviving when you realize you've mismanaged your food supplies and the caravan is months out) and a few dogs for additional defense (wardogs) and I guess emergency food (though one of my 'rules' is to never slaughter dogs unless there is literally no other means of survival, which includes foraging for plants outside). Lastly (if I have points left) I sometimes buy a few slightly better weapons just in case we meet early enemies.
A few tricks to saving points is to discard the various wood products (buckets, splints, etc) and most if not all of the cloth products (ropes, thread, etc) and get rid of the plump helmets (the actual mushrooms) as they cost four each where as other food costs 2 and the seeds cost 1. You can easily make more wood products unless you are in a woodless biome, cloth is fairly easy to make and caravans come with it all the time, and as mentioned its far cheaper to grow your own plump helmets rather then buy grown ones.
5) The most hilarious thing I've experienced is a dwarf who had both of his arms chopped* off but managed to survive (and this was in early 31, so infection was there but dealing with it was.. more difficult..), he was formally my bookkeeper but since he couldn't use his arms he just sorta wandered around. He lived until the 'end' of my fortress where a huge attack from the caverns destroyed us from within, he fought with a giant rat by kicking and biting. I had a similar thing happen recently with a warrior with nerve damage but somehow its not the same when the arms are still there, even if they are useless.
That said oftentimes I play more 'seriously' in a weird way. I mean I am amused at what happens and all, and when I tell people stories here or elsewhere I often play them for laughs.. That said in my mind I enjoy it more by being more serious with regards, its related to my imagination and how I 'perceive' the game, its a long and probably sad story to most
Good to have you on the boards, hope you have great fun with the game.
*in the interest of accuracy I believe one was pulled off.