Your asking a variety of questions, and you will get a variety of different answers because everyone uses different products/techniques. Perhaps a better approach to this would be this....follow the products instructions.
I can think of any brand of polish/compound that specifically states to leave the product on. Lets take a step back and focus on technique here. Using too much compound/polish will cause it to cake up and create a huge hassle for you once your ready to remove it. Not to mention its a waste of product. Using the right type of towel to remove the compound/polish is also very important. Finally, using a cleaner will not only aid in helping you remove compound/polish residue, but it will also give you a clear indication of the paints condition by removing oils that often mask potential swirls.
1. Use the right ammount of product.
2. Use the right microfiber towel.
3. Select a good paint cleanser.
Assuming your a professional, technique shouldnt be an issue for you, but I take it your still new to detailing, so I will fill you in. Contrary to popular belief, quality microfiber towels CAN and will scratch your paint if used incorrectly when removing compound/polish. Be gentle when removing the compound. More often than not, many detailers like to go crazy and tend to "scrub" the polish away instead of gently wiping it off. Change towels often, because the compounding/polishing granules will condense and have the potential to micro marr the paint you just spent your time perfecting.
If you follow the basic 3 steps I mentioned above, you will have no problem removing any brand compound/polish. Unless your using M105....stay away from M105