If he is willing I'd sign Brickley to a one way deal, even if he misses the NHL roster cut off, he is waived, if he passes through he is good for the AHL and as a NHL call-up.
Strome I have no idea why they would not just keep him unless a trade came up. Once his contract ends next off-season, just don't qualify him but offer him something lesser and short term. If he signs it, okay, if not let him walk as a UFA.
Vesey, Fast, are also going to be a UFAs next off-season, Not sure where that is going yet I'm not envisioning extensions all around. Parts of other deals before their contracts run out perhaps, some swap similar to that of Spooner/Strome maybe?
I don't think Andersson has the significant value to use to move up. Again maybe as part of some deal but as the prime piece or something that really moves the needle in a package, I'm not seeing it.
I kind of think the top 9 picks are just going to be off the table even if the Rangers included their own to move up. Those teams are likely to be seeing themselves as re-builders, or be fortunate to be picking there (fall in their lap sort of thing) , so I don't think any price for them is going to make sense.
Kreider plus something like the Jets 1st if Turcotte, Newhook, etc (whoever they like) are there past 9, is something I could see as plausible, should say Buffalo, Florida, Colorado have that pick. If the Rangers also need to take back a cap hit for a year or two to complete the deal, I'd go that route.