I'll address your last two points first, as those will be easier. If he wasn't a jerk and had done as Emilia asked of him in the capital, he would have had an easier time getting help. Basically everyone hated him, and looked down on Emilia even harder, because of his actions. He was also too busy stewing to pick up on anything going on around him the first time. Assuming he had to stay in the capital anyway, his healing would have gone faster because he wouldn't have used up all his strength and damaged his gate more in the fight with Julius. And when he went to try to convince Emilia to leave, he wouldn't be so far out of her trust from all the broken promises. It also wouldn't have been his fault to the same degree, he was separated from Emilia because of his own actions, so every time he failed to save her was because he caused a physical and emotional distance. This also ties in to the next point, that the deaths are his fault now and a punishment for that reason. He kept trying to fulfill his desires as quickly as possible, and this made his situation in trying to save Emilia and the village almost impossible through burning all his bridges. If he were better behaved, he would still have died once or twive, but he then would have figured out what was wrong and been able to get everyone else clear of danger quickly, and then waited for Roswal to clear out the cult(or some other allies, if Roswal doesn't care too.) This wouldn't be as good of a story, as him dealing with the guilt and improving himself accordingly is the core of Re:Zero. As for your last point, how long the gate needed is kinda left up in the air. He may have needed to stay with Felix anyway, it's unclear. If he didn't need it healed, Emilia wasn't going to let him go to the capital in the first place, because she was getting tired of him going all-out all the time.

Anyway, now for your main point. Going from the anime, it's clear Subaru loves, or at least greatly cares for, Emilia. It is a very selfish love, where he wants to do everything for her, and doesn't let her set boundaries. This kind of makes sense as a response to seeing her die multiple times, but it is also a result of his own fear of failure and inability to separate her from the version of her in his head. It's exacerbated by his separate flow of time from everyone else, but he does forget to respect her as a person in the way a lot of young people with crushes/in love do. I think that's why Rem's quiet, unselfish love is such a good counterpoint. She allows him to make mistakes and be boarish, because she loves the real him and not the delusion she might see. The inner life of Subaru will feel different based on the viewer, and I guess it would be quite different in every different version of the story, but that is most of what I got from the story. I would say that the Subaru from the capital would not allow himself to break down in front of Emilia the same way he did in the lap-pillow scene, because he was too proud at focused on looking strong in front of Emilia. He always had some of that selfishness, helping Emilia because it made him feel good even if she didn't ask for the help. His pride just allowed him to become lazy and forget the difference between heloing her and doing whatever he wanted around her(sort of a lose sense of this phrase, sorry if it doesn't make sense.)