Meat – When baking meatballs for a soup, do I have to bake until the meat is fully cooked

20 or so minutes cooking the meatballs in the sauce has always done the job for me. You want to cook them sufficiently long so that they absorb the moisture and flavour of the sauce, but not too long such that they lose their firm texture. The reverse process of the meat flavouring the sauce is also an important one.

As a side note, it is more advisable to simmer the sauce, rather than boil it, or it will likely become too thick and may even burn.

In general, the combination of initially cooking the meatballs in the oven, and then simmering in the pan is the right way to do things. This ensures that the meatballs are sufficiently cooked, do not fall apart, and have the right firmness. Worth noting is that you should pad off the grease on the meatballs before putting them in the sauce.

No recipe alteration needed.

Fried meatballs will develop more of a crust, and thus more flavour through the Maillard reaction. In any case, unless you're making tiny meatballs (or finishing their cooking in a sauce of some sort), they're going to need to be finished via baking anyway.

I would only do the baking method if I had to make an enormous quantity at once. For home cooking I always fry then bake.