Can I run low voltage undercabinet lighting wire under overhead microwave oven above the stove
As far as code is concerned yes you can.
Have seen some very nice jobs where the strips in plastic were attached behind the front framework, just enough of a space so the one that was used could not be seen unless actually looking under the cabinets. I like the coated ones so they can be wiped off, I have seen the “tapes” of lights attached exposed and wonder how long those will work especially close to a stove. So your plan is starting out well if you can mount them well and have the control switch in a good location it can be a nice application.