With regard to the vet situation I think yours handled things really badly and changing sounds like a good idea.
My Louis is fearful in some situations, usually with dogs and men that he doesn't know. He's very rarely been to the vet in his 4 years but if I have to take him there I leave him in the car when I arrive and sit in the waiting room on my own until we're called, then fetch him from the car. I know that if he has to spend time in a waiting room in close proximity to dogs and people he doesn't know he'll become very apprehensive and will then find it difficult to be calm when the vet is examining him because his stress levels will already be sky high. If it's a male vet I'll ask him to get down to ground level for a few seconds and let Louis approach him and then be given a treat. The vet might look a bit surprised at my request but it always works because Louis realises that the "strange" man isn't a threat to him.
Hopefully Murphy won't remember his bad experience, but if you think he might, be proactive and come up with a plan, maybe something like mine, to help him next time he has to go to the vet.
The reactivity between your 2 dogs sounds like jealousy. I don't really go with the dominant/submissive idea as it's not as simple as that. Relationships between dogs can be very subtle and can also change over time and vary between one situation and another. Because Murphy is still maturing and developing the relationship between your two is probably quite fluid and changing frequently. Now that he's castrated his testosterone levels will begin to drop and that in itself will mean both dogs will adjust their relationship again. It's impossible to predict what that adjustment will mean.
If the flashpoint is when you're petting them both you could try limiting that interaction to short periods for the time being and make it a calm time, not an exciting cuddle and play session that could go over the top. Watch their body language and if you see either try eyeballing the other or their bodies start to look tense just walk away from them without saying anything.
You could also try calling them both to you and get them to sit in front of you and calmly feed them a few treats each, one at a time, one dog and then the other. That could help both to realise that they can both be close and calm and nice things happen in that situation.
My 2 boys are 9 and 4 and are mostly relaxed together but are not best friends. We have the very occasional outburst of jealousy if both want to get on my lap at the same time or if I play with one and the other wants to join in. I watch out for it and if I see one eyeballing the other or any other sign of tension I get up and walk away to break the tension. If I miss the signs and they have a short spat (a few seconds of posturing and growling or maybe a 3 second "handbags" fight) I calmly call one of them into a different room for a couple of minutes to give both of them time to relax. It seems to work. I don't expect them to get on every second of every day, so I think a little bit of jealousy is perfectly normal and has to be dealt with sensibly and calmly.
Hopefully your 2 will be much the same and will settle amicably in the long run.
