+ Dunno how long to clean / kill it all -- I've quit making predictions on that; too many variables! But, if the WATER is clear now, you should be able to get the pool all cleaned up in a week.

+ Nylon brush is fine; vacuuming the debris is fine AS LONG AS THE CHLORINE IS HIGH when you do it!

+ I'd recommend starting the stabilizer AFTER the algae is gone, but if you buy dichlor, you can begin chlorinating with dichlor now -- just don't over do it.

+ Hayward C1200 will work well up to 40 GPM / 19,000 gallons.

+ Some of the most common cartridges out there are well, sub-standard, and name games are being played -- notice the second "Unicel" cartridge is NOT a Unicel! I just wrote this up for another post:
Here are 4 links to Amazon for Jandy CL460 replacements:
Pleatco or Chinese?(DOUBTFUL) => 4 PACK - Jandy CL460 Quad Pack Cartridge Filter C-7468-4 @ Amazon
Pleatco (DOUBTFUL) => Unicel 4 PACK - Jandy CL460 Quad Pack Cartridge Filter C-7468-4 @ Amazon
Genuine Unicel (OK) => Unicel C-7468 Replacement Filter Cartridge for 115 Square Foot Jandy CL460 @ Amazon
Genuine Filbur (OK) => Filbur FC-0810 Antimicrobial Replacement Filter Cartridge for Jandy CL460 @ Amazon
Check out the pictures; if your cartridge doesn't have the pleat ties . . . you got one of the low grade cartridges. My understanding is that Unicel or Filbur are OK, and of course the Hayward OEM is fine. Pleatco or no-name or private label . . . not so much.

+ Don't plan on using your softener to fill the pool, unless you've talked to your water treatment guy about it. Have you ever worked with a softner before? You'll end up doing a LOT of regenerating that way!