It seems when I'm making large pots of soup say 8 quarts, that the ground pepper sinks to the bottom & doesn't dissolve. Any thoughts?

When preparing soups or sauces traditionally you will use a sachet, which is a bundle of spices wrapped in cheesecloth, allowed to float for a period of time before you remove it and the product is finished. You could use a satchet, you can grind your peppercorns finer, and assuming you were using black, try a white peppercorn. The texture of the matured berry will be finer because it has had the outer shell removed. A spice grinder will get your peppercorns fine ground, but a mortar and pestle will turn them to dust then making it as close to water soluble as you will get.

Pepper is plant-based and can't dissolve, whereas salt is a mineral crystal and can. I also use the teaball approach often.

It's not a problem, just give it a stir before serving.

I put a small handful of whole peppercorns in a teaball and simmer that, rather than using ground.

Ground pepper doesn't dissolve and it will always sink to the bottom of the soup. But if you want it to be less noticeable, I would use finely ground pepper, or ground your own. This way it be less noticeable to eyes and tongue (in terms of feel, not flavor).