You can buy solid lather wheels, they are called power strop wheels. The issue is that they seem to come in only one size 3.5". At least I was not able to find varying sizes, and I think for 5/8 - 13/16 razors a 2.5" wheel will be optimal.
For home made wheel I am considering using pieces from leather belt for a grinder. They are 1" and that will give better control when grinding I hope.
For adhesive, I am considering double sided carpet tape, not sure how it will hold to heat, but it is a very strong adhesive.