I'm not going to address viewing order, since that's covered in this question, which I believe is where you got your list from. Instead I'm going to address your other questions.
Can you skip TOS?
I'm probably going to upset a lot of Trekkies, but in my opinion you probably can. There are a handful of guest appearances of TOS actors in later series, but I think you'll get all the background on them you need by watching the early movies. The list of appearances are1:
However, I don't recommend skipping all of TOS. Although the early episodes can be pretty dry, the last half of season 1 and nearly all of season 2, are excellent. I would at least recommend watching Space Seed, which sets up the movie Wrath of Khan, The City on the Edge of Forever, Amok Time, and The Trouble With Tribbles. Zibboz also recommends A Piece of the Action, which is definitely one of Shatner's better performances and helps motivate the Prime Directive that becomes central to so many TNG plots.
Do the new movies have any significant relationship with previous series/movies?
There are some character crossovers and references. I already mentioned Kirk, Scotty, and Chekhov appearing in Generations, but there are a few others:
- Worf's introduction in First Contact (The 8th movie) doesn't make a lot of sense unless you've seen at least the start of Season 4 of Deep Space 9
- Kate Mulgrew (Kathryn Janeway from Voyager) has a cameo in Nemesis (The 10th movie)
There are also occasional references in those movies to things that happen in DS9 (Like the Dominion Wars), and references in the shows to things that happen in the movies (What springs to mind: Worf's introduction in DS9 references the destruction of the Enterprise-D in Generations), but those shouldn't hamper your enjoyment of either.
One major exception is that the Enterprise season 2 episode Regeneration is pretty much a direct sequel to First Contact, the 8th film. The episode is perfectly serviceable if you're unfamiliar with the film, but you need to ignore some significant plot questions that the episode doesn't answer.
In general, I find that you can fit the movies in anywhere. The new movies deal with the TNG main cast, were all made after TNG was cancelled, and don't have much overlap with later series. They're pretty standalone.
I can't resist making one comment on your suggested viewing order. In my opinion, the premise of Voyager doesn't make a lot of sense without watching a good bit of DS9. Although the Maquis are introduced in TNG, their story gets a lot of treatment in DS9, and understanding the tension between the Maquis and the Federation makes watching several Voyager episodes (Especially the early ones) much less confusing.
1 There are lots more, but they're mostly TOS guest actors who reprise their roles in later series. I'm only focusing on the TOS main cast here