Morality is a complicated thing for Murphy, as he is portrayed as the morally grey character in a show that's all about sacrificing moral integrity for survival. Murphy's fairly unapologetic in being so, and has a strict priority of keeping himself alive over looking out for others or Doing The Right Thing. Up until Emori came into his life, there wasn't a single person he was willing to endanger his well being for in order to help out. Yet, as he's been growing in the last couple seasons, he's been forming opinions on what wrongs he's willing to live with, and what kind of survival he wants to have, for example, living is not enough, he wants to live but without isolation. He doesn't want to hurt people or cause people to die, but if it's him and the one person he loves over them, he will. It also helps that at some times, Doing The Right Thing also gives opportunities to help himself, and he's glad for that, typically, like stealing medicine to help the little girl in Arkadia.
War gets people killed, and thus, Murphy generally wants nothing to do with it. However, if his happiness is on the other side of that war, and he'll get only misery from where he is, he'll enter into it, but with an entirely self serving goal. If a leader makes a stupid strategic choice, you can bet your ass Murphy's going to ignore it and do his own thing. He's very, very much been a victim of war, from being a tortured prisoner several times over, to watching his friends slaughtered, to being the expendable to sacrifice For The Cause, so he considers any kind of cause other than Staying Alive in a war to be completely absurd. So, he won't want to do the Save Everyone thing, but thankfully, Emori is still on Earth, as far as he knows, and if fighting this war means protecting her, then he's all in it. But he would send everyone else's worlds to hell if the bargain was offered to get her somewhere safe and protected.
John Murphy has all of one (1) truly intimate relationship, emotionally and physically, and that would be Emori. She's really the only one he's honest with, the only one he trusts, the only one he's admitted to loving. If he were from an s4 canon point, Raven might also slide in as a close friend, but otherwise, nah. Joseph Kavinsky, while in Eudio, became Murphy's closest friend, as well as lover, and a person he trusts to a point, mostly because Kavinsky is about as self-serving as he is, if not more, and that's at least reliable. Murphy generally assumes pretty much everyone is horrible when put under enough strain, and will turn on him in a second if given the right push, so it's very difficult for him to trust and become close with people. That's not saying he can't be social - the boy never shuts the fuck up, so he has social ties with plenty of people, he just isn't particularly close with most. Also, because he's a rude piece of shit. As far as physicality goes, he can cuddle and get sexy however much is needed, as long as it's serving a purpose to him or helping him get what he wants. Which can be lots of things, or just saving his world/Emori.
Murphy will be really conflicted with missions vs just stay in the city and be safe. He'll likely eventually get on board with it, especially with Emori's safety being on the line, and once he starts getting too restless just sitting in the city by himself. It helps that Murphy's very used to a life like this - sort of living for survival and fighting the effects of an apocalypse, so it's at least something that he knows. He doesn't like being in charge or in the spotlight, so it's not likely he'll try to get that deeply involved, but so long as it doesn't hurt his own survival, he's willing to lend a hand and go out of his way for this.