You chose the worst possible faction if you want stocks and production, Kray.
Merc groups can only have (members/5)x1 stocks. So with minimum numbers, you get *1* stock. They are not corporations so much as a group of people following a leader, so ownership via stock means you only get so many. Ten members could be construed as a partnership of mercs, so you'll get two stock, meaning you have two owners (that can both move leaders in or out, move assets in or out, etc.)
As far as producing your own ships, that's not going to happen from within your group. You have to speak with others to get DCs, then you have to have them assign theirs to your factories/stations, then you need to assign the factory or station to a production group, then you have to have a hopper go to their faction and produce that item. Don't forget, also, that the mats for that entity need to be held by the producing faction and your hopper needs appropriate privs to produce. They also bear the cost of production on them, so you'll need to work out a deal to pay them to get your entity.
Merc factions are designed around what merc group are supposed to do, which is not production. If you want to know what you can do with your one stock after you create it, you can:
1) Sell it to someone to sell the faction
2) Use the stock page to change leader of faction. (If you ever are not leader yourself, such as if you actually start a production group and are sitting in that group.)
3) Use the stock to get an status overview of the faction. (When not in the faction itself.)
4) Use the stock to protect/unprotect faction assets. (When not in faction itself.)
Basically, it's redundant if you are going to remain as leader of the Merc faction, but if you don't plan to stay there, then the stock will be necessary for you.