A couple of questions about making this happen: would receive addresses first have to be generated, so that they could be included in the transactions as the spending addresses?
Yes, the hot wallet receiving addresses would have to be generated before you could create and print the print signed transactions. Otherwise, you wouldn't be able to tell the pre-signed transactions where to send the bitcoins.
Additionally, would there have to be some management to make sure that the right address had the right amount of funds before a particular pre-signed transaction were executed?
I'm not sure what you are trying to ask here.
I assumed that you would have multiple addresses in your paper wallet. Each address would hold exactly 1 BTC. Then each address would have its own pre-signed transaction transferring that 1 BTC from that paper wallet address to a hot wallet address.
Were you planning on something else?
Note: there are some people on this forum who will feel that "safely in the bank" is an oxymoron. They will tell you that you can't trust the bank or the government not to empty out your storage without your permission. For example:
http://abcnews.go.com/GMA/story?id=4832471California law used to say property was unclaimed if the rightful owner had had no contact with the business for 15 years. But during various state budget crises, the waiting period was reduced to seven years, and then five, and then three. Legislators even tried for one year. Why? Because the state wanted to use that free money