As you suggest, there are a number of factors, but the best rule of thumb I've found for pan-frying is to use just enough oil to cover the bottom of the pan by swirling it around. In general, you don't want puddles (you'll get splashed with hot oil), but if the pan isn't covered then your food might not get cooked evenly.
These days I mostly just eyeball it, and err on the generous side. Using slightly too much isn't likely to hurt your dish - unused oil just won't get absorbed - but using too little will result in under-cooked or inconsistently-cooked food and probably a lot of gunk stuck to the bottom of your pan. So if you're not sure, just add a little more than you think you absolutely need.
I would not recommend attempting to cook onions on the stove top without oil.
Another answer suggests that it's a bit like grilling; in theory, it is, but a traditional outdoor grill has a minimal contact surface. That minimizes sticking and makes it fairly difficult to burn food by contact alone. Using a dry fry pan for onions is more akin to searing, except that you generally only sear meat or fish. If you try with an onion, you are likely going to burn it before you ever get to the sweated/caramelized consistency you want.
You really have two options here. The first, and the one I highly recommend, is dry roasting in the oven. Cooking an onion involves the Maillard reaction and the dry, radiant heat of an oven is great for promoting it. Simply toss the onion on a baking sheet and roast it inside the skin for about 20 minutes at 450° F / 230° C. It will peel easily and you'll have a nice, soft, roasted onion. Don't overdo it or it will burn - the skin will help prevent this but keep an eye on it!
Your other option is to simmer or steam it, which will soften them substantially and cook some of the volatiles out (make it less pungent) but won't brown them because the Maillard reaction requires high temperatures and the boiling point of water is too low. Simmering should be obvious, just toss it in some water or stock. For steaming, either use a steamer basket or just put a small amount of liquid in a pan, heat it up to a simmer, place the onion in and cover it until cooked to your desired consistency.
Those are pretty much your only fat-free options. If you want to use a frying pan and expect to get them browned, then you'd better use at least a little bit of oil. If you're concerned about trans, saturated or polyunsaturated fats, use olive oil. If you're on an ultra-low-fat diet, use a blast of cooking spray. If you're avoiding "fried" foods out of an assumption that even a tiny amount of oil is unhealthy, I suggest consulting a dietician (if you haven't already) to validate that assumption first.
Best Answer
Use less heat.
It's not clear if you used any fat here. If you do, you have to be aware of the smoke point of whatever oil you're using.
By turning down the heat you'll decrease the amount of fat you are burning, and turning into smoke. If you want to still get a nice crisp exterior you can briefly sear your sausages on high heat prior to fully cooking. Although, his might be irrelevant since you don't really get good sears with a non-stick surface.