I think the use case is for a beginner game PC. Yeah, if you want better multi tasking, an i7 would do it. But this guy is budget constrained, and streaming, etc was never part of the requirement he set.
That being said, whenever I build a system I pony up the cash and spend it on the i7 overclockable CPU. My 4790k easily overclocks to 4.7GHz at stock voltage, and can even do 4.8 at stock. I think if I upped the juice I could hit 5GHz. I'm pretty sure this system will be more than adequate for a few more years.
The Debate between an i5 or i7. Well that realy depends on the video card you are going with basicly. Going with the 6700K will only realy make sense if you go with a higherend gpu like a GTX1080 or similar. But with a GTX1060 or RX480 it wont realy matter much.
If the OP is not sure if he even likes PC gaming I would stay close to the bottom of his price range. On the monitor, that's a hard one. One way is to go cheap. On the other hand monitors last longer then CPU's and GPU's in practical terms, ie: my 24 inch lasted 2 builds and now will on be my secondary downstairs PC. One way to go is a really nice 27 inch monitor and a cheap APU build. This is NOT for most people. It does give the OP the option of saying" Hey I really like this and save up for a killer system" or "I liked it better on my couch with my controller and big screen" Just something to think about. Also the OP might check if there is a Microcenter nearby for the in-store deals and open box stuff.
I don't disagree necessarily, but investing in the currently available AMD stuff feels.... wrong. That might be a viable option in 8 months when Zen is finally available to buy.
I hate the idea of the dual cores. I know they perform, i know. It is just back in 2006 again... Can we kill all dual core CPUs finally? You want cheap? 860K with 460 and you are done for under $500...
An i3 can run games like GTAV or Far Cry 4 without hacking or issue because it's a 4 thread CPU., an OS is going to see a 2 core hyperthreaded CPU the same as a 4 Core CPU in that they both have 4 threads