Filling in the amount on the checks is OK, its not like you are making them out to yourself.
As far as the signature part...instead of a signature you should do Signed Delivery Notice and in the remarks type "see check" if they did not leave an actual notice. Their signature on the check itself can serve for that purpose.
Under no circumstances would I ever accept a key to a storage unit or even go into one without the customer there. Too much liability if something comes up missing, too much hassle keeping track of keys if that stop gets cut from one route to the next. We dont do keys.