It depends on what youtube is willing to provide. It seems more likely to provide a thumbnail if you use a youtu.be link than a youtube.com link.
Instances running Lemmy from 0.19.4 onwards (I think) provide the option of providing a separate thumbnail manually, which is useful, but your instance is on 0.19.3.
As Admiral Pat mentions, embeds are easy enough. I don’t know how Tesseract does it, but a low-tech solution is to just replace ‘watch’ in the URL with ‘embed’ and stick it in a iframe. From Lemmy’s GitHub, it looks like there’s been work on this, but I’m not familiar enough with it to know whether it’s for future versions that haven’t been released yet.
New videos used to come in to Lemmy as expected. There’s been some regression that stopped it. It’s possible to bring them in manually though (by searching for the URL), and - like with embeds - it’s possible that it’s been fixed but not yet released.
PT’s videos channels are ActivityPub Group types like Lemmy’s communities, but it doesn’t handle federation the same way. It does it in a way that’s more compatible with Mastodon. Lemmy’s communities Announce everything they receive (posts, comments, votes, etc) and so if you receive that Announce, then as long as you trust the community, you can trust that the contents haven’t been changed and process it. PT’s video channels only Announce new posts (so on Mastodon, it appears as if the channel has Boosted content by the channel owner), but for everything else, it’s a combination of sending out a ‘post update’ (which is essentially an invitation to query the outboxes it provides for votes and comments), and just flinging out the comment as is, without the HTTP signature. If you get that comment, then you can either use the LD signature that Mastodon includes to verify, or you can look at the ID, and fetch it from it’s source. As such, Lemmy’s federation model is mostly Push-based, whereas PeerTube’s is a bit of Push, and a lot more Pull.