Levi formerly served with the Survey Corps, a military organisation, where he headed a special operations team of the best fighters that the Survey Corps had to offer. He is an extremely regimented man when it comes to his own habits. He has a way that he likes things to be done and he ensures that those in the chain of command underneath him adhere to those preferences. Back in his home world he was extremely respected for his ability to kill Titans, hailed as the best in the Survey Corps by fellow soldiers and the public.
Generally he appears to not talk very much at all, keeping his thoughts internalized, however he does have rare bouts of chattiness which can come as a surprise to those around him, even those who know him well. He is not one to come up and approach someone else for the purpose of a chat because of this and all conversations will likely be incredibly hard work, so he would probably thread best with chatty or forwards people because of this.
Levi is going to be opening up a tea shop within Wildewall, as that has always been an aspiration of his if one day they managed to defeat the Titans and he could remain on the surface with citizenship. A lot of people could likely meet him through this, if you'd like a thread in his shop just let me know. If a character wanted to become an employee that would be great too! They'd have to show they can clean a table to flawless perfection to be hired though.
Friendships are hard for Levi to come by as generally to be his friend he has to respect you first, for him to respect you he has to be given a pretty good reason.
Back home it was usually prowess in some form of military aspect, be it strategy or skill in fighting Titans, or even just dedication and bravery to the cause. Here in Gossamer getting Levi's respect might be a little harder but this angry smol really does need some friends here, however much he might say that he is content to operate on his own.
People who are dirty in their habits, clothing or personal hygiene really do not need to apply for a friendship position. Levi detests things being dirty and therefore, if you are, he isn't likely going to hang around in your company if he can help it at all. It could be fun to terrorise the guy by having someone exceptionally messy enter his tea shop and mess things around, but, be aware of the wrath you might be on the end of if you try this.
Levi has never had a romance with anyone before, it isn't something he looks for and I tend to think that he needs to form a strong emotional connection to someone before romance or intimacy even becomes something that he is interested in. Due to his aversion for uncleanliness Levi isn't likely going to be interested in sleeping around with people for no reason other than the pleasure it provides, there needs to be some substance there and even then, he probably won't initiate or indulge in that kind of thing often.
To summarise: it would have to be a really slow burn if you were interested in everything and there wouldn't be a whole lot of M threads. If any. There would be plenty of fluff and development though if people were interested. Due to the fact that the affection and romance would be born from an emotional connection I would say that romance partners could be either male or female.
The only ships I have for Levi are Hange/Levi and Petra/Levi, but I'm not going to be holding out for either of them.
We've already covered his dislike for the unclean. Other people who might fall into his enemies list are very small because Levi generally does not care enough to list anyone as an enemy for personal reasons. He hated the Titans, but they are no longer an issue in Gossamer. He hated the way in which the people in the Underground were segregated and those who endorsed it but, he can't do anything about that here.
People who are full of themselves for no good reason or, heck, even for good reason will be people he doesn't like, crowing about achievements and being overly prideful is irritating to him. Beyond that, pushy, touchy and invasive people won't be dealt with in a pleasant manner by Levi, so they would probably be individuals he would avoid.