Studio monitors are tuned to have a linear, or flat, frequency response. Most are designed for nearfield listening where the sweet spot is focused by angling in the monitors and listening only a few feet from them. Great for mixing but maybe not that interesting to listen to movies or play games. And anyone sitting outside the very narrow sweetspot will be losing a lot of the audio frequency range as they do tend to roll off the further off the sweetspot you are.
Home theater speakers tend to have somewhat of a V shaped frequency emphasis, more treble and bass. Can still be accurate but you will be better able to pick up vocals range frequencies and enjoy the explosions viscerally from a further seating distance and a much wider sweetspot. You can correct any speaker to have a flat response and good timing from multiple speakers with a room correction/DSP system like Audessey MultEQ XT.
OK, on to some configuration options.
The early generations of Rokit 8s were OK and the Gen 3s are really pretty good for low budget monitors. I would keep the Rokits solely for mixing since you are already used to them, but If you want to save some money and don't want to change anything else, you can pair the 8s with a decent subwoofer to move into home theater sound, i.e. deep bass for explosions, sub bass rumble and the like, sure to annoy anyone living below you. The Rokits do benefit quite a bit from DSP correction, you might be shocked how much.
Haven't heard them myself but the Polk LSiM703 seems an excellent speaker as reviewed and measured by Stereophile. Looks like you might want to make some distance between them and your ears due to the dispersion baffle they use. And pricing is around $750 EACH, so unless you know of a great sale, these are going to blow your $1000 budget.
Klipsch is continuing to have a really good factory sale until 8/12 -
http://www.klipsch.com/specials. To stay within your $1K budget and get an efficient home theater and gaming setup, I recommend the floorstanding R-28F speaker for $359 each on sale. You can add a R-12SW subwoofer, also on sale for the same price, but I would save my money there as the R-28Fs have good bass and would instead add some matching surround speakers like their R-15M for $199/pair as home theater requires rear speakers for surround sound. You can add a center speaker later, but I find that a 4 speaker system (with a phantom center) works fine.
These are passive speakers so you will need an AV receiver or DAC/AMP + source. An AV receiver offers the best value and tons of functionality for a lot less money than separates, so let me focus on that solution.
I highly recommend looking for Audessey MultEQ XT room EQ functionality in any AV receiver you are considering to be able to tune sound to your speaker characteristics, room acoustics and listening position.
Trying to stick within your budget, how about a DENON AVR-X1300W (7.2-Ch x 80 Watts Networking A/V Receiver) for $299 -
http://www.accessories4less.com/mak...-ch-x-80-watts-networking-a/v-receiver/1.html
The INTEGRA DTR-40.3 (7.2-Ch x 110 Watts THX Networking A/V Receiver) for $350 is also a great choice -
http://www.accessories4less.com/mak...-110-watts-thx-networking-a/v-receiver/1.html
The Denon and Integra will be fine for headphone amping but if you do want something more specialized for headphone use, I would go for a JDS Labs B-Stock Objective2+ODAC (Objective2 headphone amp with integrated ObjectiveDAC) for $239 -
https://www.jdslabs.com/mobile/products/169/objective2-odac-combo-revb-b-stock/
Lots of choices to be sure, but the more you know what you want your end result to be, the more the options become clear.