As far as the broken frame goes, what I would do is put a little glue on it, take a couple of pieces of tie wire and wrap them around the broken pieces and twist them tightly together pulling the wood closed. Nailing on the frame with the bees on it will make them mad and maybe break the comb. Zip ties could be used but I doubt they would pull up tighter than the wire.
Don't even bother with the butt headed replies, those people are just wanting to get a rise out of you.
Sounds like you will do fine, the bees don't really care if the frame is broken or not.
I will have to agree with BJ about the shape of the nuc you bought, looks like some one just pieced together a bunch of old drawn combs and frames. Not trying to down you here, but next time you buy a nuc or established hive make an inspection of the bees first, frame by frame. If the seller refuses to let you do that walk away.
Hang in there!!