You purchase a Triton test kit (BRS sells them). $49 for one. You will get a box with two sealable water vials and bar codes that stick on each vial. You mail the box and water samples back to the address inside the kit (may be an address label already filled out) You have to pay postage, so expect another $2 to $3. They will receive the sample and forward to the lab for the testing. Your results will be uploaded and linked to your bar code. There's a website where you register and enter your tank information. You enter your barcode information on the website. Thus, if you do multiple tests periodically, you just enter the new test bar code and the water sample results will link them up. It's not a hard process and is explained in the kit. Takes about 2 weeks or so to get results back. I think it would be worth exploring given the lengths you are going to just to rule out some things.
As far as temperature, it sounds like you're covered there.
As far as your pumps being new, you're probably right and it's probably not an issue, but it's a good idea to always visually inspect equipment periodically (when you clean them for example).
There's so many things that I've read about happening to people and their tanks. I've heard of people who accidentally dropped a screw down in their sand bed (unknowingly) and it cause issues in their tanks. I've heard of kids throwing a penny in the sump. I've heard of a cleaning lady spraying cleaning stuff near the tank and the mist getting in the water. Need I go on?
I think better pictures under whiter lights will give everyone a better idea. It's possible your corals were already a little stressed in your nano tank (by your own admission). Moving them to the new tank probably stressed them, and given the tank is fairly new, it's just probably not as stable as we all like to think it is. Like I said, I think there are a lot of biological processes going on in our tanks that we probably don't fully understand...I think a lot of these get worked out or reach equilibrium states only after several months of a tank being established. Even in my new 6' tank I started about 2 years ago, I noticed an establishment period that took many months, despite me using existing live rock and other bio media from my existing (and running) 75 gallon tank at the time. I didn't have to cycle the new tank, and my water parameters all measured good every time I measured, but I can tell you that the tank just wasn't as stable as it was many more months later. I can't tell you exactly what changed, and this is purely anectodotal, but that was my experience. I just mention this because it seems like you are doing all the right things.