I think you have answered your questions about mental health in your first paragraph?
It's ok to feel isolated, fed up and screwed into the ground because you have no connections to others, in a strange place and not happy at all. It sounds a difficult place to be at, and sometimes the harder you try the worse it gets. That isn't necessarily mental health it is just feeling very unhappy and isolated due to lack of friends. That would be a normal feeling I'm sure for so many of us given the circumstances.
The information we don't see here is the extent of your family support, or if you have any family friend close to you with whom you can chat to express how you feel? I would see if there is a welfare section at Uni and go and see what support you can find and is available for you to help you find new people and groups or even to be signposted to talking support services. You won't be alone feeling that way, there will be hundreds of you all in the same boat. Try the social section of the student union too to find new outlets for your spare time too, often they need volunteers to do hundreds of jobs. Arranging freshers week is the big one.
One thing is for sure. No one will come to dig you out of your room and in your room you are isolated. Nothing will change if you don't make it change. Start by understanding what makes you tick, what you like, dislike, sports, leisure, socialising, etc etc. What have you never tried but would like to? Make a list of things you would want to do, then go and see if you can find someone else who would want to do that? Try and find some team games where clubs are looking for individuals who want others to join them?
If you rant online what are you ranting on about? Are you angry and offloading or just expressing an opinion? If you get drunk what is your relationship with alcohol. Alcohol just numbs everything and stops the pain (for a short time) Why are you drinking? But you can have control over what you want to do about that? You have a choice to live your life with your head wasted and wasting time on this earth or you can make the most of every second of the day, being alive, using your senses to explore the world. Plan to learn something new every day. Seek out new experiences every single day. Be determined to make change happen, and it will. Is the drinking a bad habit or do you think the problem is much deeper and you need to get professional help to quit. If this is the case get help, but beware of putting yourself on the NHS database if you don't really need the help. An NHS problem drinker label can stop you doing so much in your later chosen career.
Often you will never know what you are capable of in life until you have tried it? Many times emotional feelings are painful but perfectly normal, and they don't need medicalising, just surviving. Sometimes people get stuck on a life event and live their life continually through that prism with long term adverse effects because no one has ever shown them how to cope and deal with life's really awful situations. Bad events need acknowledging but then realise that the feelings and the state of mind doesn't last forever. Life moves on. You will move on. But you have to create your own life. No one will come along and do that for you? You are in charge of that now and you have the opportunity to create your own luck and your own potential.
Keep smiling. Don't take life too seriously. Keep looking forwards not back in life. Plan today what you will do tomorrow. One small piece at a time. If you have lived, and you are still upright and breathing you are doing very well. Wake up and live again tomorrow - Get going and never waste a day.