Hi Camo & welcome. I agree with others, and your thread title - you've definitely jumped in at the deepest end! As well as asking for advice here, and the (sporadic?) help you have available, I'd be seriously looking for some on-hand support, of experienced horse people & trainers. If you have to pay for that, so be it. Matter of fact, considering the situation, I'd suggest sending Mo off for some basic training first.
And to be gelded. Aside from other reasons people have given for gelding, the fact that you're already in the deep end without knowing how to swim... continuing with that analogy, having 2 stud colts and a mare is like being in that situation, with crocodiles to boot!
Mo is very rambunctious and wants out of his stall but he is also barely bridal broken.
He doesn't yet need to be trained in a bridle, but I'm guessing halter trained is what you mean? I'd work on(or find someone to do it) getting him halter trained now then. Should only take a couple of days to teach him the basics.
Is he stalled full time?? Then I'd send him to somewhere that he can be out in a paddock with other horses, until your fencing is adequate to let him out at your place, with a dividing fence between the others. Aside from disease quarantine management, I'd put him out with the others ASAP. He can learn how to be sociable with the other colt, while Mum ensures good behaviour is upheld. I don't foresee a big problem with that... assuming you're taking the testosterone out of the equation.
2. how should I introduce them so there are no issues/fighting
You won't ensure there are no fights. That's how horses establish their 'pecking order' and it is likely the mare will have to put the new boy in his place. It's not common for them to hurt eachother badly though. Being young boys, the colts will also likely play fight quite a bit. As others have said, introduce them first with a good, safe(pref timber) fence in between, for maybe a few days, then, so long as they have an open paddock, room to move & stay away from eachother, not get cornered... put him out with the others & keep an eye on them for the first time.
3. as I live in a cold environment I was thinking about keeping them in stalls for the winter and only putting them out to pasture on nicer days.
No, you can't do that. Well, you can, but it's seriously not good for horses to be cooped up without regular exercise. Mentally it's also not healthy for them. There are many health & soundness aspects that are effected by cooping horses in stables. Instead, I'd keep them out 24/7 but have stables/shelter open for them to go into when wanted. If you must stable them, do it for as little time possible daily - perhaps overnight, and ensure they're out for at least a fair few hours daily. In addition, if they're not going to get much exercise on their own, due to weather or otherwise, take them out for on lead walks or such daily.
5. should i feed them alfalfa or hay or a mix and should their diet include a mix of oats cracked corn and such as a supplement
Grass hay is a good basic diet. They need enough that they will not go hungry for periods of time, if you're not feeding free choice. If the baby isn't weaned, Mum will likely need a lot more calories. Feeding some alfalfa along with the hay will provide some extra calories for Mum. A small amount of oats mixed with alfalfa chaff or such could be good, *so long as* you can feed over at least 3 meals daily. Otherwise I'd avoid grain & stick to a 'mare & foal' premix, which should also include necessary nutritional supps, which will otherwise need to be added. Youngsters may do fine just on hay and a nutritional 'ration balancer' type supp for foals.
6. I am placing a salt and mineral block in each stall, is that good or bad
I'd just put a plain salt block down, and supp minerals otherwise, as explained above, as they don't get much at all from blocks.
7. I want to build a coral for working with my colts and training, what is a good size, should I build normal wood fence or a portable metal coral?
Depends on your situation & funds what type may be best, but size-wise, you want it big enough to do all the basics in, so about 8m square or round would be a good size to start them in, IMO.
I have a 5-6 acre pasture which should support the 3 easily. I also have a 10+- acre hay field that I plan on fencing in to allow for grazing after the second cutting next year.
Ditto to others, but if you also have the 10 acres & can make about 4 paddocks out of all & rotate them, or strip graze around the outside(track setup, look up 'paddock paradise') and save the middle for pasture/hay(& play space), that might be adequate.
You will want to find a good vet asap - good to look for one & register with them ASAP, before there's a need. You will also need to find a good farrier, to get their feet attended to every 6 weeks on average. Sounds like the babies need training before they're up for that though. It's your responsibility to have them well trained enough for the farrier, and the sooner the better, as I gather they've not been seen to yet and it's far easier to correct minor problems now, than wait for them to become major ones in a more mature horse. Horses also need to be seen by an equine dentist about yearly on average. Often more frequently for youngsters.