absolutely you can - just want to make sure you use the right prep and materials - your grout is thicker than most so it won't be quite as smoothly finished but I think anything is better than the brown (lol on your sweater comment)

here is someone on YouTube who paints her shower (in another video she paints and stencils her floor) https://www.youtube.com/watch?v=VkaGTyAAQ1A

I don't think the tile is as ugly as the grout. I would get a grout dye (made specifically for that) and dye it dark so it doesn't show as much as it does now. It will blend in better with the tile. eta: try it out behind your bottles to see what you think.

seal the grout and live with it. doing anything will be a waste of time and make it look worse.

Go ahead to get estimates from pro tilers to re-tile the tub surround and the floor and from professional bathtub and tile reglazers. Then you will be able to compare costs in view of the years before you replace the tile and tub. In my area, reglazing is sometimes used when worn tile has to be covered for a few years. It doesn't last decades, but it often looks and works just fine for several years.

Super helpful, thank you!