What is the minimum and maximum amount of salt to water ratio to make a brine?
This is a very complicated question--in the chemical sense, salt dissolved in water is a brine, but they can vary in concentration (usually measured in percentage of salt by weight).
For the culinary purpose of seasoning and helping meat cuts stay moist, it varies considerably; the stronger the brine, the shorter the brining time, in general, although below a certain threshold you are going have little effect. Too much salt in the brine, and it will get to the point where you are are wet curing, and the product will start to take on hammy textures and flavors.
A good range for culinary brines is about 3-5%, depending on the length of time you intend to brine, and the size of the cuts being brined.
Does brine have to be boiled? I've seen some brine recipes that call for the solution to be brought to a boil, then cool. Can't I just stir the solution until the salt and water are fully incorporated?
No, you don't have to boil the brine; doing so is for ease in dissolving the salt (and possibly sugar).
For safety, you only want to brine in cold brine, so you would then need to cool a previously boiled brine. This is usually done by boiling half the water weight with the salt/sugar, and then adding the other half of the weight of water as ice after.
Some brine solutions add ingredients other than salt. When does such a solution become a marinade instead of a brine? And what's the difference?
Marinade is not defined scientifically, so there is no formal distinction. It is a matter of culinary purpose. My personal take is that brines are made of water, salt, and sugar, and possibly one or two flavorants; marinades often have an oil, dairy, or acid (such as orange juice or vinegar) base, although there is considerable variation.
Marinades are about adding flavor (usually), and possibly tenderizing (when acid, or an enzymatic actor such as pineapple juice is present). Marinades act at the surface of the food product.
If the primary purpose is to help the meat stay juicy, and to season it, the application is brining, so call it brine. Brines, with enough time, will affect deep into the meet, although there is some argument over what the exact scientific mechanism is.
The thing is, despite the myth, brining does not help flavor (as opposed to salt, for seasoning) enter the meat--adding extra ingredients to it beyond salt, sugar, and water is not generally helpful, as these flavors will not pass the cell walls (they are too large at the molecular scale), so they don't actually have much if any effect on the meat other than at the very surface.
Here is a link to an article from Virtual Weber Bullet with great information on brining, and a myriad links to even more resources.