You may not have purchased/unpacked a bed yet. Not having a bed prevents you from being able to sleep in your apartment. As for the cheating thing, it's not a bug.
The long and short of it is if you hang out/go on dates with other people, even if you didn't fuck them the rumor can make its way back to your partner. If the rumors get strong enough, your partner might believe them and think you're cheating even if you're not. Since it's the Best Friend, they'll always start with the Open Relationship inclination and give you the option of becoming open rather than just breaking up with you.