Dependant on how the fuse is set up then a bad or non contacting fuse could cause the alternator not to charge the battery.

The way a core charge works is if you bring in the core (ie the part you are replacing) then you do not pay the core charge. If you do not bring in the core at purchase you pay the core charge, then when you bring the core in at a later date, you are refunded the orginal core charge.