Yes, the lower retainer is meant to be permanent (doesn't come out) and is cemented and runs along the back of the lower teeth - specifically in two places - the lower cuspids on each side. You do not see it from the front but it makes flossing those teeth even after your braces removed very difficult, hence the water flosser. It can be and should be done from time to time, but you need to use these flossing leads that look like a needle to get the floss through the incisors and under the retainer. Its a big PITA lol so I rarely did it.
To maintain muscle health, I chew[ed] sugar free gum all the time LOL. It also helps keep your teeth clean & obviously helps with freshening ones breath too. But yeah, despite not using any retainers for a long time now, my teeth really haven't moved too much, they spread out a bit I suppose, but you cant tell visually. I notice it when I do floss though. Its a lot easier to get the floss in between my teeth now, especially the lower incisors.
I think mouth breathing is pretty common. I slip into mouth breathing from time to time, especially if I'm sick and cant breath through my nose well. I always sleep with a water bottle on my side table, so if/when I wake up, I drink some water to combat dry mouth. My Dad uses those nose strip things to help open up his nasal passages to reduce mouth breathing.