The graphics card isn't going to effect the monitor.... It's going to effect gaming performance. The biggest thing is how many monitors are you going to be gaming on? If you're going to be gaming soley on that monitor then a 7850 would most likely run every game today at 60FPS+ with max settings at 192x1080 resolution. I'd say the 7870 is going to be a good choice.
yes, a new card would be recommended if you were using less than 1080P before this and struggling for frames, personally, i have a 7850, and its awesome (albiet it is overclocked), so a 7870 would definately be enough
The performance boost over the 9800gt will be incredible with the 7870, it will handle 1080p on the monitor easily. However, you have to remember that it will also use more power over the 9800gt so make sure you've got a strong enough power supply.
Have a look at your cpu, if thats outdated then you may have a bottleneck on the 7870
a 7870 is a really good card for modern games, but if you want something more but not for a lot of money, try to get a 7870 LE which is a 7800 card only by name and actually uses the GPU used in 7900 cards.
I am not sure why everyone says or downplays 1080p monitors its not that easy to max out. I have oc msi gtx 680 at 1212 core clock and I run ultra everything maxed out except msaa on some games. Example farcry 3 ultra mssa is only a 4 so my fps stay at 40 if I go to 8 kinda lags at ave 30fps at 16 the games unplayable. Is the game maxed out yes but not really not every setting truely. I have 23 in samsung led 1080p. It is easier then anything higher then 1080 obv like 2 monitors but not as easy as everyone states. Especialy with games like crysis 3 I can run ultra but garentee I can not completely max it out and be playable.
I will give you the most sound advice there is I myself researched this for 8 mo. the best card out there for the money and the same in price as the 7950 is Asus Direct2 CU II any of the 3 versions different clock speeds is all. but all about 300 bucks but important to get asus if you want the best of the bunch. it even out performs the gtx 580 it has same chip as 670 and 680 the gk 104 chip all they did was reduce it to 192 bit.And it is 20% percent cooler and vastly quieter then anything out there trust me on this or research it and u will see.