Is the snaffle a single or double jointed?
A single jointed loose ring, should leave 1cm on both sides between the rings and the horses mouth, while at resting position. As you take contact with a single jointed snaffle, the bit shortens in the mouth and the rings therefore come closer to the lips and side of the face. If the bit is too small, it will pinch. Too big, and it will slide across the mouth, not only causing discomfort, but also 'dulling' the rider's aids.
A french snaffle (double jointed), you want the rings to be sitting just next to the horse's lips. When you take up a contact, the double joint action means that the rings will move evenly from the lips. If you have 1cm between the lips and rings, the bit is too big and will be ineffective and uncomfortable.
As for bit thickness, it does sound like yours is too thick. It depends on the horse as to the thickness of the bit, if he has a fleshy mouth or low pallet, he will be uncomfortable in a thick bit, so go for one a little thinner. It's really a case of trial and error.
For a bridle, obviously you want an english style bridle. But otherwise, whatever he is happiest in. A standard cavesson noseband, possibly with a flash attachment depending on his needs, will suffice. There's no 'fancy dressage bridle' that will magically turn him into a dressage horse ;)
The saddle, yes a dressage saddle WILL help your position and effectiveness of your aids. A GP/AP or jumping saddle will only inhibit your ability to apply the subtle and precise aids required in dressage. They place your lower leg forward into a slight chair seat, the seat is quite flat and overall, making you fight to hold your position while you ride.
However remember, there are no 'one type fits all' with saddles in terms of both horse AND rider. No matter how much adjusting a saddler does, some saddles will just not fit some horses. And again, one rider may love riding in a certain saddle, the next may hate it to the point where they become physically sore riding in one. For example, the Isabelle Worth line of dressage saddles, in particular the old models (the new ones are quite nice), are so unsuitable for me, that even after a 10 minute ride in one, I am in so much pain through my hips, knees and lower back, from being literally forced into a position and not able to move, that I have to get off. However I have friends that absolutely love their Isabelle and ride well in them.
You will have to try a few different saddles to find one that suits both you and your horse. Certainly not a case of popping into a saddlery or onto Ebay
and buying the cheapest or 'prettiest' saddle as chances are, it won't be the 'right' saddle for you and the horse.