This got long. Pardon me. I had more stuff to air out than I thought.
It depends on where you are and who you surround yourself with how much the place is Irish or British aligned. Essentially that identity marker is what a lot of tension comes down to from my experience. Although it's usually coded as Catholic or Irish. It's a lot better but there's still a heavy Catholic (Irish)/Protestant (British) divide. The town I live in feels to me more like the Republic because I'm near the border.
You can cause a lot of offence if you call someone Irish or British or even Northern Irish and that's not what they call themselves. Bonus tidbit: If someone asks your last name in Northern Ireland, they're really asking are you Catholic or Protestant.
Full disclosure: I don't consider myself British. Honestly, it gets my heckles up a bit when I've been called it and I don't really know why. It's deep and instinctual and while I'm not patriotic at all, being Irish is a big part of me. Factually I'll accept I'm "technically" British but that's as far as it goes. I'd call myself Northern Irish or European before British. Given the choice I went for an Irish passport and when talking to some people (like when visiting the states), say I'm from Ireland because fuck explaining Northern Ireland's existence.
I'm aware my background informs my feelings a lot. Both my parents were born and raised down south. Da, Catholic, from County Monaghan and Mum, Protestant, from County Cavan. They moved up north after having my older brother. My siblings and I were all raised Catholic and went to Catholic schools. My mum actually wanted my other brother to be Protestant but that would mean he would have had to go to a different school because you need a baptismal certificate to go to a Catholic school. A lot of my extended family live down south and especially when I was pre-teen we were back and forth visiting a lot.
Even considering I said things are better, just over ten years ago I would walk through parts of my town in fear while wearing my school uniform because it informed others of my religious identity. I'd get sworn at, I'd be threatened, I'd get stuff thrown at me. And people on the other side got this too. Having parents who in a mixed marriage, I was so confused about why this would happen. I didn't get the link between Catholic/Protestant and Irish/British identity and other politics that came with that for a long time. Honestly I think a lot of the people (generally teenagers my age and slightly older) also didn't really get why they were doing it either. Plus I was realising I was an atheist then so that added to identity confusion.
To actually answer your first question bearing in mind how my experience informs it:
Northern Ireland is basically part of the UK... With a lot of the Republic's baggage. We are both part of the UK and Ireland and part of neither. We are the leftovers of what England could cling on to for a foreseeable future and what Ireland decided were worth sacrificing for getting back most of what they already had in the first place. Both doing so in order to get some amount of temporary peace. We've not got out of either place's shadow enough to fully establish our own culture but I think we've made something pretty interesting any way.
For the second question:
Before Brexit I would have said a flat no. Sentimentally it would be nice but the Republic doesn't have the NHS and... That's about it now that Brexit is in the picture.
Northern Ireland's main deal is agriculture and most of the money to sustain that came from the EU... After London sent it to them. We get no help from London directly. We're simultaneously left to do our own thing but get none of the resources to better ourselves. We simply chug along.
I believe being part of the EU would be more beneficial to us than being part of non-EU UK.
It's still scary to say yes though. The transition would be messy. I've been fortunate in the sense that I was born after the worst of the Troubles and was too young to process what was left after. I was seven when the Good Friday agreement was signed. (Fun fact: The DUP was the only major party that opposed it.) It would be scary and feel like a big unknown but I'm starting to think it'd be worth going for.
The Republic is even working on a form of national healthcare. It's slow and a mess but it's trying. The Pound is struggling enough we'll probably do grand on the Euro. I think it's more Northern Ireland that wants to go back than the Republic wanting us back. We have more to gain from it than they do in a pragmatic sense. It'd do wonders for our sense of being wronged by the English though. Be worth sharing a pint.