Actually I have run 1440p on what is very close to your current system. On a 5930k before I overclocked it (so 3.5ghz) and with a gtx 970 4gb (factory overclocked) and I ran 2560x1600 (above 1440p) at 60hz.
Which 1440p monitor to get depends on your budget.
This one is on sale for a remarkably low $310
https://www.amazon.com/Philips-BDM3270QP2-LED-Lit-monitor-2560x1440/dp/B01LXPX9I4/
It is 31.5" and 60hz. Best text quality/clarity of any monitor at any price that I have seen.
This one is IPS (better colors and viewing angles) and 60hz and costs 260.
https://www.amazon.com/BenQ-GW2765HT-27-Inch-2560x1440-Monitor/dp/B00KYCSRSG/
This one is a TN panel (faster, but worse colors and viewing angles), 75hz and costs 295
https://www.amazon.com/ASUS-PB277Q-WQHD-2560x1440-Monitor/dp/B01EN3Z7QQ/
This one is a 27" TN panel with g-sync and has a max refresh rate of 144hz. It is on sale for a very cheap 430
https://www.amazon.com/Dell-S2417DG-24-Inch-LED-Lit-Monitor/dp/B0149QBOF0/
This one is possibly the best monitor right now. It is 1440p 32" 144hz and costs an eye-watering $850
https://www.newegg.com/Product/Product.aspx?Item=N82E16824025836
If you want to go above 60hz, your PC will need significant updates. If you just drop in a 1070 on a 144hz monitor, you will not keep up. You will need g-sync or to use medium-high settings at most. A 1070ti is around 20-50 more for a similar quality model. While a gtx 1080 is around 20-50 more than a 1070 TI and faster. Which one depends on your budget. The gtx 1080 is sort of the sweet spot on frames per $. Below the 1070, and you might as well stay with the very capable card you have now.
Ideally you should have 16gb of ram. I would also upgrade the cooler and overclock the PC some. Or at least add some case fans and try with the 212.
Try to overclock your video card a little with afterburner would be a good thing to try. If you are going to push your low-end PSU with a bunch of overclocking.
https://linustechtips.com/main/topic/631048-psu-tier-list-updated/
My actual advice is this:
Settle on how much you feel like spending on a 1440p monitor.
Think about if you care enough about faster than 60hz to spend extra on video cards, etc.
But the monitor. Try it with what you have. Overclock the CPU a little. Overclock your video card some. Consider buying a slightly better cooler if you are CPU bound in your games and your 212 cooler won't let you hit the speeds you need. Consider adding 16gb (2x8gb) of ram OR getting real good and keeping background tasks (like chrome) closed while you game.
If after trying it with what you have, then you will have an idea what to replace next. Hopefully you didn't over spend on the monitor so you have the option of other upgrades IF you actually need them.