I believe that non-professional military can get moods, but once they become permanent military they're immune. I've never had a champion get a mood, and wrestlers will hit legendary within a season at best.
As for your questions:
1. You can't have them carry two different weapons or dual wield in this version, or next version for that matter. If you're worried about weapons sticking, go with hammers. Blunt weapons have 0 stick chance, and if they get a nervous wound while training they cross train into marksdwarves well (crossbows use hammerdwarf skill in melee). Maces are also a viable choice, but I wouldn't recommend them. They do less damage and don't cross train as well.
Spears main advantage is their stab damage, which can even the odds against a large creature due to a high chance of a critical wound. Unless you have a HFS, you probably don't need speardwarves unless you just want them.
2. I've never had problems with military dwarves tantrumming from a dead baby shield. However, I had a husband become miserable for a season once, til she gave birth to a new baby shield. If you're worried about it though, I've never had a marksdwarf even get hit after being properly cross training (legendary wrestling, then optional hammerdwarf training). Leave them with leather armor and bucklers, they'll be fine.
Wrestling: I'd train them to legendary+5 wrestler and beyond. It's not hard to change their weapon preferences, what's likely happening is they stripped an article of clothing off their sparring partner and didn't drop it. Currently, if they're holding multiple items in their weapon hand they wrestle instead of using melee weapons. You can get them to drop that by setting them to wear no armor, then after they drop it all reassigning them to the armor you want. If that doesn't work, check the wiki on Wrestler, covers all the other methods.
Fishing is not sustainable for a large fort, but you might be able to manage it for a small fort. I'd recommend setting up farms though. Even small farms can produce absurd amounts of plump helmets with a skilled grower. You can also check to see if there's wildlife, a good hunter can provide very large amounts of meat, leather, bone, and fat (processable into tallow, which can be cooked into food) until the zone gets overhunted.
Haulers: up to you. I don't, but they gain a lot of skill doing the unskilled labor in my fort. Most of them are up to professional in furnace operating, milling, threshing, butchering, tanning, and are at least *no adjective* growers from helping with the harvest (note: don't set "All Dwarves Harvest" until your grower is sufficiently skilled, mine's past Legendary+5 so it's just providing a buffer of farmers in case he dies). Pumps are an easy way to train them without building relationships to start a tantrum spiral.