Your wallet version is up to date. Your wallet is telling you it can't reach the daemon.
To see why the demon fails to start, check the $HOME/.bitmonero/bitmonero.log, you should have information about any error at the end of it. You might have to add "--log-level 1" in the wallet settings page as daemon parameters to get more verbosity in the ...
No. Unless they have some kind of access to your local machine of course.
The issue main issue with using a remote node is the operator can log (and associate), your IP address to your transaction(s). That and you're trusting they serve up valid blockchain data.
Can this be accomplished with a large USB-Stick
...and decent write speed?
Define "decent". USB 3.1 gen 2 is pretty fast. There are loads of USB drives on the market so you'll need to research carefully which you get.
Current blockchain size is about 73GB...
No, it's more like ~90 GB unpruned and more like ~30 GB pruned. Hence you can ...
The cache contains data that's retrieved from the blockchain. It includes the hashes of blocks (used to detect reorgs), then data for every transaction that you sent or received, custom information like user tx notes, secret transaction keys, that kind of thing. It will grow as you transact in Monero.