Test chlorine and pH every day or at least every two days. The other tests a couple of times a month is usually fine. The exception to that would be when you are tweaking something like cya or alk levels. Then, you'll need to test more often until you have it where you want it.
I don't know much about cartridge filters other than the fact that most people think they are a pain in the rear end. But, a 1.5hp pump is way overkill for a 13,000 gallon pool. My pool is the same size as yours. My original pump was a 1hp but when I replaced it about 4 years ago, I asked Al (Poconos) our resident pump expert to recommend something. He picked out a 1/2hp full-rated inground pump for me to buy for my above ground pool and it has been plenty for my pool. Bigger is not better when it comes to pool pumps.
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