Hey! fellow (sort of) goth here. I couldn't tell you much about the courses to be honest, but I can offer my views on the cities though! :-)
I have only been to Durham a few times, but I have seen a (let's say) "alternative" scene there, unfortunately there are a lot of people who don't understand that culture. For example, I was walking across durham bridge (the old timey one) and saw a guy in dreads playing metallicas 'Kill em all'. He was getting weird looks from the local chava, but I was encouraged that there was a scene there, there are plenty of shops that give off a "whitby shop" vibe. There were at least two cool shops doing new rocks and docs. As far as the nightlife, I couldnt say, but the presence of alternative people there was encouraging.
For Birmingham, it is probably the least of my experience. I go there maybe once a year, but never had time to sample the nightlife. I will say to watch yourself on the streets of Brum, my family is there and they have told me to be careful on the streets at night (I have visible piercings and usually wear black), obviously this advice is true for wherever I go, but I feel I should extend the same to you as I have heard stories from good old brum. but please definitely look into it in the absence of any real experiences I have had there, I do not want to discourage you from somewhere that might be best for you
Now, Leeds is where my largest experience lies. I love Leeds, pure and simple. There are downsides to every place of course, but I really enjoy the diversity there. The only unfortunate thing is that there arent alot of rock bars there (only two I think) and those two rock bars are rather expensive (unlike, say York, which has a few rock bars with good drinks offers) There is a massive amount of diversity in Leeds I havent seen anywhere else in the North. There is also a good LGBT scene in Leeds if that is your persuasion, though the same could be true of Brum and Durham.
To end this ramble, Ill just list off a little bit of what I know in a pros cons style thing. This is by no means an exhaustive account of all the cities, just what I know
Birmingham: Pros - Big big city, lots of choice for bars/clubs and social settings, lots of cool shops and some really cheap housing.
Cons - Probably the one citiy of the three that I wouldnt feel comfortable walking around at night (Birmingham is the least of my experience so take these points with generous piles of salt!)
Durham Pros - Amazing country surrounding the town, old style of city reminiscent of Whitby or York, there is a scene there, but may need some research as to where is good to find that culture as it was hard for me to find a good rock bar. Great shops and markets, selling nu rocks and docs as well as some awesome "nichey" shops. Some great museums there too, they have a great oriental museum too
it's one of those towns where I seem to run into some really nice people, Like cardiff in that respect.
Cons - Lots of Chavs (not necessarily a bad thing sometimes!) Town is quite hilly so if you have mobility issues it may be a pain after a while, the scene may be hard to find, but at least there is one!
And, Leeds:
Pros - Diverse as all hell! walk through a predominantly muslim neighbourhood and grab some great schwarma, turn the corner and you could be in a student area, round another corner you'll be in an upmarket area. Good LGBT scene, great diverse culture with some great museums and art galleries (some really great homegrown places hidden away too, look for them and youll find some amazing art scenes) Roundhay park (go there on a sunny day, you're welcome
Cons - not many rock bars, and the ones that are there are quite expensive, youre better off going to the bars that have offers to tank up then head to the rock bars (like the shed) for a drink or two, Severe homeless problem if your heartstrings are easily pulled, rent can be a little above average, though if you go to a student area, it can be quite cheap.
Now my mind has gone blank, I am sure Ive missed stuff, but feel free to ask me any questions. Im no expert, but Ill do my best to offer what I do know :-) any specific questions, go ahead