If you are only going to be here five days and this is your first time, forget about non-tourist stuff, you won't have enough time cuz you will need five days to do the tourist stuff! Do the tourist stuff first. You've had many good suggestions. In addition to the highline, visit the Whitney Museum at the southern end of it. might walk across the Brooklyn Bridge from Manhattan, take a tour of the historic Brooklyn Heights when you get to Brooklyn, then head back across the Manhattan Bridge, which will leave you in the heart of Chinatown. This will be about four miles of walking, so make sure you can walk like a New Yorker. lol.
There's also the top of 1 WTC and the Empire State Building and the 9/11 memorial. If after that if you are still in the mood you can check out the new high end shopping at Brookfield Place.
It's my fourth time in NYC within six years, so hence me asking for non-tourists stuff as I've seen all the musts. Sorry, forgot to mention this at the beginning [emoji4]