I make THREE gallon batches, not six. I try to avoid dipping the mop in the bucket to keep the water clean. I’ll rinse off a dirty mop under water first, if I can, and wring/shake out the water.
No need to rinse it unless it looks dirty or you get sand in it. The sleeves should be washed with your towels. Never hurts to launder them. Some do it less often than others, but I have quite a few mop covers, so the dirty towels from the job AND the mop , get tossed in the washer.
I use small mesh, laundry type bags for all my dirty towels, pads, and mops. Keeps the dirty stuff organized rather than it getting everywhere or dropping out the door and having to bend over and pick it up.
If you run across outside windows that are THAT dirty to mess your mop up every window/every other, you are better off taking a quick swipe with a soft brush to knock all the dirt off BEFORE you mop up the window. Doing resi, you’ll likely need to do that anyway for the cob webs and leaves.
Pick up a brush like this Libman bench brush. Works great. https://doitbest-weblinc.netdna-ssl.com/product_images/601965/601965/5638bf8f69702d4ba80a290d/large_thumb.jpg?c=1521285490