The rating 2600mAh (or 2.6Ah) means the battery will produce 2600mA for one hour, or 1000mA for 2.6 hours of indeed 1mA for 2600 hours. The rating is the current multiplied by the time the battery can produce that current. In practice the rating depends on the current. The figure of 2600mAh will have been obtained for whatever current is optimal for that battery. Running at higher or lower current will give a shorter lifetime.

Incidentally, the 9.62Wh rating is the current times the voltage times the time. So in this case the voltage times the current times the lifetime is 9.62Wh. If the battery voltage $V$ was constant the 9.62Wh would just be the 2.6Ah times $V$ (so presumably the battery voltage is around 3.7V). However the battery voltage changes as the battery discharges, so the relationship between the Wh and Ah ratings is a bit more complicated than this.

As for your questions about charging, it's impossible to comment without knowing exactly what sort of battery it is. If it is a 3.7V battery that sounds like a lithium ion battery, in which case you may well have cooked it by connecting it to a 12V PSU. Lithium batteries need a special charger.

It might be worth you asking on the Electrical Engineering SE as they tend to be more interested in the nitty gritty of anything electrical. But if you do post there be sure to give as many details about the battery as possible.