How to prevent my roux from sticking to the bottom of the pan?
There's already a question about roux/stock temperature (as well as a recent question that's probably a duplicate). Summary: yes, adding cold liquid to hot roux is good. Additionally, most of what I'm saying is already in the answers to "How to mix a roux with stock". I've gone ahead and tried to address your specific concerns, but I really think thoroughly reading rumstcho's answer there would help you.

Yes, it's also bad if the pan is too hot, and you rapidly cook the roux and boil off excess liquid, leaving something that was briefly wet enough to spread along the bottom of the pan, then a second later dry enough to cook onto the bottom of the pan. This is also one of the reasons it's bad to add too little liquid at once. If it ever dries out, you've messed up.

And yes, you should probably be adding more liquid initially, at the very least enough to wet all of the roux. It's fine if the first batch of liquid gets you something substantially thicker than your end goal, but you shouldn't be just trying to get a slightly wet ball of roux, then a slightly wetter one, and ten steps later have a thick slurry. You want to give yourself enough liquid to work with right away.

Once you've got all that, you need to whisk well. If anything is remaining sitting on the bottom of the pan the entire time, of course it's going to stick.

With respect to your other questions: Spices shouldn't hurt you, and a roux containing significant onions and garlic isn't a normal roux, but I've certainly made tons of thick sauces with all kinds of things mixed into the roux, and had no trouble. Bits of things are just, well, extra bits of things. They don't really interact with the liquid when you add it.