Does my monitor support 2560x1600 res?

Here is the screen I'm using RIGHT NOW: http://h10010.www1.hp.com/wwpc/ca/e...4302-215012-215012-3770740-5034316.html?dnr=1

Here is the graphics card I'm looking at getting: http://www.ebay.com/itm/EVGA-NVIDIA...CS%2BUFI%2BIIUM&otn=2&pmod=370586074809&ps=54

Will my monitor boost up to that res. or even 1920x1200 with this GPU?

Well, as you know, the native resolution is 1920 X 1080. All the other details as to maximum should be on the HP site. What you list is for HP Canada. Pay close attention to contrast ratios as well.
The video card goes to 2040 X 1536, but I would never depend on a used graphics card to do what you want at those resolutions.
Short answer: should work, but I would want to see it in the details on the HP site.

I'm not buying that card. just that model. I'm getting it new, I'm sorry, but does "native" basically mean max, or what? How would I find what the monitor can withstand? I have one right here.

Native in this case just means standard resolution. The monitor will always want to reset to that res.