Can I store the whole blockchain yet keep my actual wallet with bitcoins offline? I want to have the full blockchain "just in case" yet have my wallet be OFF the internet.  Like, on a laptop that's just for bitcoin, with wireless turned off.  Or possibly on an external drive that is backed up on multiple other external drives.  So can I use bitcoin core, and just store the blockchain there.  And use my other wallet when I want to receive money.  I have a substantial amount of money in coinbase and I don't trust keeping it there.
Yes you can, 
https://bitcoin.org/en/developer-reference#importaddress

What you need to do is to create a watch-only address on your online computer.
This can be done by "importaddress" in console and use no rescan. Rescan, after you have imported all the addresses to get their respective balances.

To spend BTC from your offline computer, you need to sign the raw-transaction offline( created on your online computer).
Take a look here how-to: https://bitcointalk.org/index.php?topic=651344.msg7306076#msg7306076

If you use Electrum wallet, all these steps can be simplified a lot.