Yes I should check that one out. I've had success using Sync to sync other dynamic files that I need to keep up to date but again, it can be a little clunky and it mostly works at the file level rather than random bits of text that aren't even technically at the level of a file (I don't think?)
You can share the clipboard. I always do it between mobile and Linux pc. I assume you can't keep the history of what you copied into the clipboard in Mac or iOS.
What security issues are you talking about? Universal Clipboard (via Handoff) operates over BLE locally. iCloud is used in part of the process to facilitate key exchange.
If the government/other actor is going through the effort of getting Apple to somehow disclose this key (assuming that it would even be possible) and lurk outside your window, you have bigger things to worry about.
Read the link I provided. Apple documents the security mechanisms of how a lot of their services work. Universal Clipboard (and handoff) does not use iMessage.
I thought Continuity was an iCloud thing, rather than a bluetooth thing. I have concerns about iCloud and iMessage but bluetooth I'm seemingly more comfortable with