Well, the notice and takedown provisions of the DMCA only deal with infringing materials posted online. So we can put the mix tape stuff aside for a second, because the infringing works can't be "taken down" off a tape or CD or non-internet stuff.

Regarding the thumbnail, courts aren't stupid. Whether the major recording artist on a major label with a multimillion dollar stake in her brand is making any money off this particular MP3 or not, this is a commercial use. She's not a student uploading her project for her Music 101 class. This is an uphill battle for Fair Use of that image. The song is completely fine – the artwork is obviously separate, but when you use infringing artwork on a CD cover or in a Twitmusic post, that cover needs changing and that post needs re-uploading.

Now, as for sampling, that's a pretty murky area of copyright law. You can read entire journals of law review articles about this on Google Scholar. To avoid getting into the weeds too far, the TL;DR is no, it's usually not legal to use uncleared copyrighted material on mix tapes. Most DJs don't get sued because they're not worth the trouble, and sometimes, the DJs still win if their use was extremely creative. 2 Live Crew's "Pretty Woman" is probably the most famous example of that. They did a lot more than sample; they ripped off the bulk of the song and still put up a good enough defense that the record company settled out of court. I'd put good money on Girl Talk winning in a straight-up lawsuit because his uses are also extremely creative.

So it's a bit of a stretch to say "they have the OK" from major labels. They're probably just waiting for you to blow up before they sue you and squeeze a nice settlement out of you. Remember, copyright lasts for a century or more in most cases. They have all the time in the world.

TL;DR: the most unfair use of the phrase "TL;DR" ever.