1) 3000MHz (with two sticks in Dual Channel) is a good idea. How much BENEFIT there is depends on many factors. In many games for example, having 2x4GB 2400MHz would give sufficient bandwidth to the CPU to not be a memory bottleneck.
There will be some situations where you gain as much as 15% vs a 2400MHz memory kit from my understanding. So it's not a HUGE deal, and with a budget build you should factor in the cost.
2) Motherboard compatibility (AFAIK) is the main issue not the chosen CPU in determining which kit of DDR4 memory works.
3) I believe even with the "exact" same DDR4 kit some use different memory modules. Some work at full speed (i.e. 2933MHZ) and some have issues going above 2133MHz.
4) BIOS updates have helped, but...