The NTS path on this game focuses on the MC enjoying how his girlfriend being depraved. The NTS choices are when MC will ask Liz to be more daring, like wearing more revealing clothes, flirt with other guys, post lewd pics on ShareSpace, etc. Maybe in future updates the MC will even ask Liz to have sex with others, while chatting him to give update on what's happening.
However, the cheating with Matt still exists. The only difference is that in NTR-NTS she is cheating almost everyday while in HappyNTS the cheating only happens on certain circumstances, like when Liz is at work or you selected a bad choice that will give chance for the cheating to happen.
For your other request, see below. Enjoy.
Cenor
NosyFellow
NTR
NTRS
Happy
Lovers
SkillIssue
Gangbang
AFR POL SOLI
Angel
Devil
Cute
Lewd
exhibitionist
silhouette
sleepover
punishment
back view
wet
Hidden in plain view
beach
anticipation
alternate
I think I am missing a code or 2. I'll reply back once I find them.
Also, these codes are only until v0.20, but be aware that v0.21 is already out. You can visit Patreon or other sites with the latest version like F95zone. And since it's a new version, there are new codes in there provided by Nosy when he message you at the end of each path. I did not include them here since you are not yet in the updated version. So update your version and get those codes when Nosy chats you in the end.