First of all, thank you for rescuing this boy!
I agree - another vet visit is in order. Just like doctors, not all vets are as thorough as others, degrees of competency, time constraints, etc. all figure in. We worked with one vet for quite some time. He knew all the cats we brought in were ferals, and unless we reminded him every time, he would forget to test for FeLV AND for parasites!!!!!!
As Jenny pointed out, not eating hard food is often a sign of gum disease or tooth problems. Again, another good reason for a second vet visit.
Re: the throwing up. This could be several things. Round worms often cause cats to vomit (they're often called "tickle me" worms), although you can usually see them in the vomit if that was the cause. They look like long thin round worms - sometimes coiled up. It could also be that he was not used to the diet. Cats in general have very sensitive stomachs, and new types of food often cause upset stomachs. Switching food should be done slowly over at least several days and up to a week, gradually increasing the proportion of new food to old food.
I'd get his teeth checked to see if that's the problem with eating dry food. It's SO MUCH easier to leave out dry food to let the cat free feed and then feed him wet food once or twice a day - more like a treat than his main source of nutrition.
Our cats never finish their wet food.
I don't know how often you feed him, but think of cats living in the wild. They eat small rodents and sometimes birds and often what garbage they can salvage - but because of this, they eat many small meals very frequently. Unlike humans, cats are not geared to eating just two or three meals a day. This is very likely why he doesn't finish his food and yet complains because he's hungry.
About not drinking water. It could be that he's getting enough moisture from the wet food. However, many cats prefer drinking moving water. Our newest addition, Ming Loy, will not drink from a bowl - only from the Drinkwell Fountains we have. There are two very common water fountains for cats, one made by PetMate and one made by Drinkwell. We've used both, and all of our cats prefer the Drinkwell fountain. It actually "pours" the water out into the catchment bowl. The PetMate has a slope like a slide that the water runs down, so while it creates movement in the water, there is no free-pouring water. Either can be easily ordered online at Petsmart.com.
The grooming issue - I don't know anything about this. Perhaps pose the question in the Care and Grooming forum where you'll get people more familiar with that specific issue.
Re: poop covering. If you see him poop, as he leaves the box, hold him gently and take one of his paws and help him scoop litter over the poop. This has worked for us several times when one of the cats starts leaving poop uncovered. If it doesn't work, I don't know what else to suggest. It does take some time though. FYI - even if he does start covering his poop, it's still best to scoop the litterbox twice a day. Cats can be very particular about litter box cleanliness, and it's so much easier to keep the boxes clean than to deal with a problem of not using the boxes.
Re: being overly affectionate and meowing constantly. Do you work and are gone throughout the day or are you at home? Long absences can contribute to this problem, and the thing that can help (even if you are at home) is play. Play with him for 10 mintues in the morning - really active play with a wand-type toy - and for 10 minutes at night (at least). Also, doing things like hiding treats around the house for him to discover before you leave for work can help keep him active and interested. The constant need for attention can be that he's bored. Perhaps consider purchasing a window seat for him (or a cat tree that is at window height) and buy bird feeders that you put on the window with suction cups (or if you have screens, something you can attach next to the window - or if on the first floor, they have large hangers you can stake into the ground and hang feeders off of). This is total "cat TV." All of our cats LOVE watching the birds and squirrels that the feeders attract.
And though it's difficult to tell, perhaps he could use a pal? Sometimes another cat as a companion works - though this would really be more for if there's no one home during the day. If you decide to try this, please read up on new cat introductions. Cats are very territorial, and they rarely like "new pals" right away. They have to get used to the idea of each other's presence first, so it's best to keep the new intro separated in a separate room. We have to help them associate the each others' smell with good things, etc. - and then by the time they actually get to meet, the introduction usually goes pretty well. There are many threads about it, and there is an article about new intros in either the Behavior or Care section of the part of TCS that isn't the forums (tabs up at the top of the page).
As to the meowing - some cats are talkers, and there's not a lot to do.
Re: the allergies. If you have insurance, you may want to take a quick trip to the doctor. I am highly allergic to cats but successfully live with six of them. The thing that helps me the most is my prescription for Zyrtec-D. Frequent vacuuming REALLY helps. When we moved into the house, we bought a Dyson vacuum cleaner, and though they are very expensive, I cannot tell you the difference it has made. They are GREAT for people with allergies. And if you don't already do this, start vacuuming your furniture every time you vacuum the carpets. You can also consider HEPA air filters for several rooms - especially the bedroom, and near where you sleep. Make sure you dust the intake areas frequently and change the filters according to instructions, if not more frequently. You can also add Febreeze Allergan reducing fabric spray to your shopping list, and spray your curtains and furniture frequently.
Good luck, and thanks for caring for this lucky guy!