I use the showerhead as a water pik (but I shower after eating before work).
You should use mouthwash before brushing, as brushing is actually about rubbing toothpaste into the tiny crevices in your teeth, forming a protective layer to prevent bacteria and other shit from getting in there. You should brush before you eat, and then rinse your mouth with water after you eat. Chewing sugarfree gum helps as well (and helps stimulate saliva production and your saliva naturally deal with bacteria).
No, the alcohol in Listerine is actually working against that as it dries your mouth out which helps the bacteria grow. Your saliva actually is a natural deterrent to bacteria. And STFU about them having some secret formula. They literally have to put the active ingredient on the damn bottle (
https://www.drugs.com/otc/113880/listerine-antiseptic.html). And guess what, its not alcohol. That's just there for the people that don't know alcohol is bad for mouth health but have associated the burn from it as such. Also, alcohol consumption is tied to about ~30% increase in mouth and throat cancers.
And uh, having the gums contract is actually very bad for your mouth health, as it exposes more of your teeth to things that cause its decay. Which, frankly a rinse of Listerine isn't doing that, although multiple daily of it might cause your gums to recede which wouldn't be a good thing (and would be happening because the alcohol is drying your gums out, not keeping them healthy). That's how you get gingivitis.