Budget battery pack - safe to charge?
Will the protection circuitry keep it protected while charging?
Most likely, although I would be very cautious with it.  It would be a good idea to be 100% sure it has protection circuitry.  If the cells have protection circuitry then nothing else is needed, if not, then other protection circuitry is definitely needed.

Will the charger work? 
Yes, it should. But if it charges at 1A then it will most likely put unnecessary strain on the batteries and shorten their life. Typically you want to charge batteries at less than 1C (which for those 880mAh cells would equate to 880mA). I know there are cells out there that can support much higher rates without major impact to cycle life, however most cells are not made for that.

Is that charge time reasonable?
Yes, 2-2.5hrs is a perfectly reasonable charge time.  This is due to the fact that it charges slower once it hits the voltage limit.