They're obviously going to have to cut stuff. It's annoying, but at least we know up front.
No they're not! They might have had to make cuts, they might not. You don't know, I don't know. Until there's either an official announcement or someone gets hold of the actual discs and watches them, speculation is pointless.
The music rights for the songs used in Season Two have *nothing* to do with the rights for the songs used in Season One - or any other Muppet Show, for that matter. Rights clearance is, as someone's already pointed out, a complex business, and things change all the time.
For example, it hasn't been possible to clear Beatles recordings for any purpose for the last few years. It's got nothing to do with the money, the recordings are not available at any price. This wasn't always the case, and it may change again in the future. (Before someone comments that the Muppets covered a number of Beatles songs, I'm talking about the actual Beatles performances here, not cover versions.)
Five years ago, "You've Got A Friend" was evidently clearable for DVD, because the Vincent Price TMS was released in at least three countries with the song intact. Two years ago, either it was no longer clearable, or else Disney opted not to pay the asking price. (A cover of the song was being used on a TV commercial in the UK around the same time, so it was obviously available for at least some uses, but again DVD release doesn't necessarily have to have been one of them.)
It's also worth noting that rights usually have to be negotiated for each country the DVD is to be released in, and a song that's clearable for (say) the US isn't automatically cleared for the UK (or vice versa). The US release of 'A Muppet Family Christmas' is missing several songs due to clearance problems; the UK release includes these songs.
The fact that some Season One shows were cut for release is disappointing, but has no bearing on Season Two.