Maybe in poetry, but in day-to-day life it would sound rather odd.

Yes, "of" can be omitted. No, it doesn't change the meaning. Omitting it sounds a little old-fashioned and literary: The first thing that popped into my head was a line from William Shakespeare's famous play, "Macbeth:" "Macbeth! Macbeth! Macbeth! beware Macduff; Beware the thane of Fife. Dismiss me. Enough."

Yes, you can say that. "Of" is understood even if you don't say it. Look up the famous nonsense poem "Jabberwocky" by Lewis Caroll. It contains these beautiful lines: “Beware the Jabberwock, my son! The jaws that bite, the claws that catch! Beware the Jubjub bird, and shun The frumious Bandersnatch!” Some of the words are nonsense words so don't expect to find them in a dictionary.

Both official and unofficial notices and guidance usually include ‘of’ - ‘Beware of the dog’ for example, which householders put on their gate, or ‘Beware of ticket touts’ outside entertainment venues.

Just to add: "Beware! Anyone who says they are looking to exchange money are probably trying to scam you." And "Beware of anyone who says they are looking to exchange money, they're probably tying to scam you." Are two different ways of structuring the phrase and what might sound like a simple omission really isn't. The first phrase sounds a lot more emphatic than the second.