To answer as best as I can
For your budget running things at high shouldn't be a problem for some time to come. Air cooling is the simplest and you can have air cooled and reasonable quiet too. GPU wise a non-blower card is your best bet - yes they vent the hot air in the case but with reasonable case airflow you shouldn't have any problems. CPU cooler wise - this is a bit trickier. I would honestly say find a case you like, fits where you want it, has enough hdd bays, optical bays etc. before picking your cooler. Some of the best CPU coolers out there are massive and it does limit your case choices.
Twin displays is always good, I use a TV a second monitor and it handy to have around when I want to watch tv while doing PC things. You'll probably find your graphics card will have an HDMI our but probably more display ports. It will be your monitor that you'll have to check the spec of more though I think most decent ones will come with at least a display port and one HDMI.
With monitors I'd say there's several things to consider;
1) Resolution: This is the same as with TV's though there's more resolutions for monitors especially if you're considering ultrawides. Generally I'd probably say stick to 1080p (1920 x 1080) monitors or QHD / 1440p (2560 x 1440) monitors. You can increase upto 4k monitros (3840 x 2160) but honestly I don't think this is worth at the moment as we're really only now getting GPU's powerful enough to run games at decent frames at 4k.
2) Refresh rate / FPS / Hz - Both the same thing. I think even cheap monitors run at 60Hz which is what you want at minimum though if you're considering you might want to play shooters then you might want to consider 120Hz or 144Hz monitors.
3) Panel type: Different panel types have different strenghts and weaknesses. As a generalisation IPS has better colours and wide viewing angles, TN have higher refresh rates but smaller viewing angles. There's other things to consider to like VA panels, LCD, LED etc. but the best thing you can do is have a rought idea of what monitor you want and read reviews.
As for size, if you're going to be at a desk I'd say you're probably looking at the 24" - 30" range unless you're consider ultrawides.
Honeslty I'm not sure there's a wrong choice right now as Intel and AMD are pretty equal these days. There's somethings Intel will be better at but things AMD will be better at too. Are you only planning to game on the PC? WIll you be doing anything else? If you're not doing multithreaded workloads then it pushes you towards intel but with AMD's resurgance we're starting to see more games go multithreaded. Plus AMD have said they will keep the AM4 socket until 2020 (Ryzen 400 / Zen2+) so potentially gives a small upgrade path whereas Intel change sockets more often. Though as above, if you're not in a massive rush I'd consider waiting to see what Ryzen 3000 / Zen2 is like as it's apparently releasing Q2 this year.
As for overclocking - turbo boost technology is pretty damn good these days with CPU's boosting close to their maximum clock speeds automatically. If you're chasing every last % you will get better performance manually overclocking but the difference between automatic clock boosting and manual overclocking is smaller these days.
Finally on RAM - again depends on if your pc will be used for other things. If you're only/mostly gaming then I would say 16Gb is all you need for now. As long as you're not going small form factor then most motherboard will have 4 RAM slots and 16Gb most commonly comes as 2 x 8Gb so if you ever need more you could.
Yes just a case of where you plug you're usb. I have mine plugged in to the front of my ps4 so it is quick and easy to unplug and in to the front of my PC.