It should work, but I would get a new power supply, as the power supplies supplied by HP are specifically made to fit that case. Not that the power supply would not fit in that case, its the fact that the cables are only long enough to reach where they are supposed to.
16GB is pointless for any application btw, especially gaming. Go for 8GB. That case is sorta unneeded huge as well. If that's what you want, go for it.
It should all fit, being that HP generally sticks to proper form factor for their motherboards.
I am aware that the build you have now works (I thought you had no purchased the RAM though, my misreading)
The thing about the power supply, is that if you look in the case, they only made the cables really long enough to reach the parts on the motherboard from where they are. Therefore, if you get a new case, then the cables would not reach, as the power supply is in a different location.