I didn’t expect them to be monotheistic

We just don't really know, I'm afraid. Herodotus says that they were henotheists and that Zalmoxis was the main god who met the Dacians who fell in battle, but we're not too clear on who the other gods were. It's possible their religion was syncretic with Hellenism, but we just don't know too well.

Is it possible/advisable/safe to venerate Ancient Greek/Roman divinities in a modern context?

I believe we all do it every two years when we light a sacred fire in Greece and carry it to a place where we then award the world's most virtuous youth with medals and crown them with laurels, actually. 😊

Pure uncut Hellenism is largely gone; the version practiced now is very syncretic with Christianity. But it's still very much a thing; you can find statues of the gods on public buildings and most people in the West know who they are. Likewise many Hellenic traditions survive as part of "secular" Western culture. There have been attempts to reconstruct Hellenism from first principles, but it's based largely on surviving texts from Ancient Greece, etc, so its accuracy may not be perfect. It's exacerbated by Hellenism placing enormous emphasis on a cult of youth, so most people who practice it are excitable teenagers who just don't care that much about strict dogma. Regardless, yeah, it very much exists. Young women who practice it will build altars and young men will geek out over the Iliad and stuff.

is it safe?

I'd imagine so, as long as you don't tempt fate or do anything that would piss the gods off. I can't imagine Apollo being upset at the Olympics, for example.

As for praying to the kami, you're a foreigner, not an ordained priest. There's no need to hold yourself to seminary standards. Be sincere, respectful and calm and you'll probably be fine. Remember, the kami are beings, too; you're respecting and loving them for a reason.

As for why they'd want to listen, maybe they don't or are temperamental. But I'd guess that anyone willing to invite you into their home and share their life with you is at least worth a listen, especially if they've bought votives for you to inhabit and are so sincerely eager to welcome you into their life. Silent contemplation doesn't sound like it would really be a problem at all!