For ambidextrous imo you should go for Steelseries Sensei, Zowie AM-GS or Razer Taipan. I'm a FPS gamer myself. I'm using the Steelseries Sensei (Fnatic edition) and it's serving very well indeed. My aim has become much better. In the other hand if yu're determined about getting a Logitech, get the G400. It's much better than the G300 and even it beats the G500. And of-course yu'll need a good surface to get the 100% from any of them.
For keyboard I hope I'm not too late to say this but Logitech G110 and the G510 provide a very bad quality of Anti-ghosting and I'm saying this from my own experience. Get the X4 Sidewinder. Trust me, you won't regret. That's the best you got within this budget. One of my friends bought one two months ago and as he says it's a really the best one you can get.