As long as you have a 'smart' NiMH charger (which this Sony BC-CS2A appears to be), you can use it to recharge any brand of NiMH cells. 'Samrt' means that the charger monitors each cell's voltage individually , and stops charging when the cell is full. A 2500mAh cell will take 20% longer time to become fully charged than a 2100mAh cell, that's the only difference.
While I don't own these specific batteries, I do use other similar NiMH batteries with my X360 controller. I find that they do last a long time; I can't tell you exactly how long, since it seems to vary by game. My rough estimate is approximately 20 hours. It's definitely a cost-effective alternative to regular alkalines and Microsoft's proprietary rechargeable battery pack.
Because NiMH batteries are 1.2V (rather than 1.5V for alkalines), the battery is never fully charged according to the Xbox battery meter. I usually start out with one bar gone on a fully charged cell. Just don't let that make you think they're defective.
My recommendation would be to use regular AA's on your infrequently used controller, though, since NiMH batteries leak charge faster. You will probably not get 20 hours out of them if it's spread out over 20 weeks.
Yes, you can charge batteries with a charger made by some other companies as long as the type of charger matches the type of the batteries. For example, if the charger is for NiMH batteries, you can only charge NiMH batteries in it.