+ Yes, leave it out while killing algae. It's not going to live algae well, and it will stop up and interfere with circulation.
+ TSP = tri-sodium phosphate, an industrial detergent (no suds) good at removing pool goo from filters.
+ Yes, rotate the filters. You can save the solution, and use it multiple times. So long as you only put a little bleach in the mix, you can leave the dirty filter in the mix, till you're ready to finish cleaning it and using it to replace the cartridge that in the filter. You might want to get something like a Rubbermaid Brute trash can and lid, and simply leave if filled with cleaning mix and cartridge all season. The link goes to Amazon, but it will probably be cheaper if you can find it locally, especially since you have to buy the lid separately.