After using latex paint and cleaning the brush/roller with soapy water, how do I dispose of the paint/water mixture?
First I brush out most remaining paint, sometimes I use old newspaper or old cardboard. When the brush is no longer leaving paint on the scrap paper I squeeze the brush between some absorbent paper a few times to squeeze out any remaining paint.
Only then do I wash the brush under running water in a utility-room sink, in which case the very dilute mix goes down the drain.
Paintbrushes are so cheap nowadays that sometimes I don't bother washing them (particularly for brushes that have been used with oil-based paints). I just buy new ones :)
For paintbrushes I am planning to use the next day, with the same paint, I find that wrapping them in clingfilm can sometimes prevent the paint drying in the brush and keep them usable without the effort of cleaning them. In this case it helps to have more paint on the brush.
You want to clean your brush / rollers in running water as far as possible, say in a sink or tub. I would probably clean the brush very well and simply change rollers between paint colors. If you need to roll the same color another day, just seal your roller in a 1-gallon ziploc bag or similar (make sure it's airtight), and you can use this easily enough even a few days in the future. I've read that you should freeze the roller you're planning on reusing, but have no personal experience with that option.
Again, this is only with latex paints. I'll defer to someone else for advice with oil-based paints.
Disclaimer: Ziploc or any other plastic bag, as long as it's airtight, won't matter. No endorsement suggested or implied.