i'm sorry you had such a rough intro to mini horses. The experience I've had with biggies and Shetland ponies that are somewhat larger than most minis (on this forum??) is vastly different...
For pills - you put it in the syringe, hold your finger over the tip and add water. Maybe 20CC (hopefully you have a syringe that is that large at least). Then put the plunger back in and tip it up and press out some of the air to allow you to set the syringe in a tall cup or vase. W/I 1/2 hour or so, pill will be liquefied. May take longer - depends on the pill. Hold your little guy's head up - my smallest mini (35" at withers) I can tuck his head thru my right arm at the elbow, my hand is under his jaw on the right side, I face the same way as the horse, standing on the left. You just put the tip of the syringe into the corner of his mouth, angle it back about 1/2 way up the syringe and keeping his head up, push plunger. If he's not letting you (jumping around), you can tie him so that his head is held short and high (about the same as your physical hold), then as soon as you've given him his meds, you untie him. Don't let him go right away - hold his head and make sure he's swallowed. If you want you could use honey or juice or applesauce to the pill. I've treated many horses and we don't usually have time for that kind of thing (all of those take time and are MESSY BUT they DO make the horse happy and look forward to his "treat"). Be careful with them though as they could all affect laminitis or founder - I don't know amounts.
Ask 10 people for feeding advice - you'll get 10 different answers. I think you'll do best following YOUR vet's and YOUR feed store's advice - with knowledge gained here on this forum. Feeding not only varies by the size of horse, but also what area of the country you are in and what you have available to feed - both in forage and in concentrates (grain) and supplements.
I feed completely different then what's already been explained and our sandy soil doesn't support lush pasture grass in the past 10 years, so ours are out year round. No grazing muzzles. I grew up feeding horses and ponies (many were mini sized) corn, oats and bran and had some of the best looking animals around w/ no supplements - in the mid-west and CO, WY, and MT. Different pasture and different hay. During the winter - breeders often turned their youngstock or broodmares out on the cornfields - with no issues. Just a different way of feeding. I still know quite a few breeders (and some are pony breeders) who do this today.
Your guy was hit by a lot of new things at once - a large amount of feed (corn is considered a "HOT" feed), new grass pasture (different from years of their previous owners'), vaccines, worming and TRANSPORT to a new facility. Any one of those by themselves could cause an upset with the 3 biggies - colic, founder or laminitis. Just can happen. And some folk are just lucky and never see any issues at all, no matter what they do.
For other info - there are several mini "stores" on line thru this board (see above) and I also have linked quite a number of stores on my links page. There are quite a number of mini horse books that apply to just minis - really neat! Also, there is at least one company that offers videos - for training as well as care. Giddy up Flix dot com. There used to be at least one other - I can't remember the name ... if you do some google searches you may find it!
I'm very happy for you that your new pen is built (lucky you to have a hubby who can/did do that!!) and that your vet came back out. Soo sorry that this happened. You did a great job taking care of your guys - just keep looking after them. They are cute! I think you'll do just fine!