How bad is Latex Paint down drain and how to remove it?
Latex paint uses water as it's thinner. It's water soluble and won't dry as long as it's wet. If you wash your brushes and rollers with plenty of running water you should be fine. I clean mine with the sprayer and even then it takes quite a while to remove all of the paint. You can also set your brush in a bowl of water and let water run over it for a few minutes to thoroughly flush it.

If you dumped it down the sink, you might have a problem but should be OK if you flushed it down with plenty of water. Hopefully your problem is just in the p-trap, which you can easily remove for cleaning or replacement. I'd remove the trap, clean if possible and replace if not, and see how your sink runs. If it's still slow you might need to get the drain camera'ed to see where and what is blocking it up.

As far as disposing unused paint, either leave it until your county/city/whatever has a free haz-mat drop-off, or add kitty litter to it to dry it out before throwing it away.