Is it safe/okay to water the paints straight from the bottle, f.e.: Put the water into the paint bottle, shake, and then just leave it as is?
Certainly, and even suggested. The paints will sometimes dry out, especially if you have owned them for a long time, and each paint you buy will have a different consistancy, some are a lot thicker than others. Nothing wrong with putting more water in the pot. If you use a thinner I probably wouldn't put it right in the pot in case you ever want some unthinned paint, and be careful not to thin it too much because you can always thin paint further but you generally can't thicken it up after its too thin unless you buy a bunch more paint.