Report: tiny bit of dye in the bowl after several hours, so yeah, there's a leak. I didn't hear the water running leak in this time period.
"chain holding the flap isn't kinking, rusting out" - chain is new, not kinked, looked ok when I watched it flush
"(1) turn off the water supply, (2) flush, (3) wipe the surface the flapper closes against with a rag or paper towel. See if you wipe off a bunch of alge or debris. This alge or debris could be getting in the way and causing the toilet to leak, even if the flapper is new." - just cleaned that off, will see if it helps. It was slimy.
"But first, try backing out the little screw on the top of your toilet valve so that it's not constantly pressing down (or up, depending on design) against the valve. If this adjustment is too tight, there's no slack in the system and tiny vibrations or fluctuations could trigger the valve. There should be a tiny bit of wiggle room where the float arm doesn't immediately actuate the valve." - I do not understand much of that, but there is a little bit of wiggle room before the float arm moves the flapper. I think we're using different terms and I don't know your set.
Water pressure - as far as I know, it's not excessive. I don't see any of the symptoms of increased pressure, and this is something that I have had to tackle in the house. Toilet does not run if other water is used.
"perhaps you have your water infill valve for the toilet throttled down. This directly controls the speed of which your toilet fills from a long time to a very short time. If the leak of symptom number one is bad enough, this could also increase your fill time, especially if the fill valve is throttled down." - Valve at the wall (to turn off water) is open fully. Fill valve in the toilet, I haven't messed with that and I wasn't the one who replaced it. It does have an adjustment thingy with +and - on it, but since I don't understand the fill valve in general.....
Will let this sit for a few hours, see how it does.