My setup : Motor- Sunnysky x2212 980kv
              ESC- SimonK 30A
              Prop- 1045
I intend to use an 11.1v 2200mah 25c lipo. Can I use this?

Yes you can.

Also, I read somewhere that I should have a lipo voltage checker on flight, so I went ahead and bought a LIPO BATTERY VOLTAGE TESTER AND LOW VOLTAGE BUZZER ALARM. How exactly do I use it and how will I know when the battery is low and I should land the quad?

Just connect the alarm to the LiPo balance port. The alarm itself will have instructions on connecting it. There will be button on the alarm to set the alarm value.
I also suggest using a battery with a higher C rating. I used a 2200 mAh 25C WolfPack with the same motors and it died within a month. It can only supply 55 Amps and your max draw is around 52A which is too close for comfort

So which battery should I use? Should I get 2200mah 35c or 2800mah 30c 3 cell lipos?

Depends whether your quad can lift the weight of the 2800 and if the battery will fit on the quad. 2200 35C can supply 77A and 2800 30C can supply 84A. 2800 will give you slightly more flight time.

Or what about 1850mah 40C 11.1v ?

with 1850 mah ....you may get even lesser fligh time than 2200mah

So I'm going with 2200 mah 35C.

i would suggest to go for a 3300-4200 mah 20 c    battery ....so that you can get somewhat good flight time...