Something like:
Gigabytes Z370 AORUS Gaming 5 or Ultra Gaming.
Asus Prime Z370-A, STRIX Z370-E Gaming or F Gaming.
No, DDR3 RAM and DDR4 are not compatible. Maybe buy 16Gb of DDR4 for your new pc, see how it run.
Arctic Freezer 7 PRO Rev.2: The TDP is 115w, the max cooling is 150w. On paper, that look ok. Try it.
8600K or drop your girl and get a 8700K.