I've never heard of that being a case but it should function like a regular monitor and you just wouldn't be able to use its AMD feature. If you use displayport, you can use the monitor's full potential 144hz and all just without adaptive sync.
I know you didn't mention g-sync. I just didn't want to leave it out of the picture since I mentioned the others.
That freesync monitor that I linked will work fine then. Using HDMI, you can get 120hz. Going the displayport route will get you the full 144hz. I couldn't find any other monitors at that price range with hdmi and/or displayport.
Once you go 60+fps, you really can't go back lol
After a little bit of googling, I don't think you're going to be able to get that kind of price for the monitor you want. That may be possible if you get a used/refurbished one. The closest thing I can find is a freesync monitor that has displayport and it's about $17 CAD over budget.
Freesync: https://www.amazon.com/AOC-G2460PF-24-Inch-Gaming-Monitor/dp/B01BV1XBEI/ref=sr_1_5?tag=amazon0606-20&s=pc&ie=UTF8&qid=1461901857&sr=1-5&keywords=144hz+monitor
A regular 1080p144hz is going to cost a bit more and with g-sync a lot more.
Why are you exclusive to using HDMI or displayport? HDMI 2.0 only supports up to 120hz. Displayport is probably the way to go though.
You could maybe consider upgrading to the 6700k for $100 more if you're seriously planning on streaming and video editing but other than that, it's lookin' good. Or add an SSD like Remmi2002 said.
This monitor is able to natively overclock to 165hz and I was wondering if it was safe to keep it at this rate without damaging or greatly decreasing its life span?
This is out of your budget range by about $100 but I can personally vouch for it's quality. It's the Acer Predator XB271HU. It's 27", 1440p, has G-Sync, has an IPS panel, and 144hz but overclockable to 165hz to name the main features.