In the beginning part: Liz does not want to go to work. If you choose to force her to go to work, the path will become Happy. Afterwards, when Liz ask to hangout after buying new lingerie, if you refuse her, the path will become NTR.
And no, from the beginning until the latest version, Sam only teases the MC but never have sex. At least, not yet. We will need to wait if that will happen on future updates.
Lover is simply agreeing on everything Liz wants. If she doesn't want to go to work, just be ok with it. Spend time with her, visit her, have sex with her, etc.
NTS is short for Netorase where a guy gets horny in seeing his girlfriend being depraved. Typically asking Liz to show extreme lewdness and be more vulgar. You need to ask her to wear more revealing clothes, show off and flirt with other guys, post lewd pics, etc.
Come to think of it, I remember the Ashani drawing of Sam having sex with 3guys. That may happen with any of the possibilities below:
1. It might be a shooting for 4some on her work.
2. (This will probably answer your question) Since Liz will go to the outer branch and will stay for 1week, MC cannot go because of his work. Sam then decides to go to Liz then she comfronts Matt. Sam might be fucked by Matt and 2 other guys (probably the same guys who fucked the purple-haired MILF). Most likely Sam will agree to fuck them in exchange for Matt to stay away from Liz. This will be OK for Sam because she is already used to big dicks in her shoots so I'm sure she won't get depraved and be influenced by Matt. Also, I don't think she has nanobots that can affect her emotions just like Liz.
