I raised my own dd as a single mum from a similar age without any real support network at first too - you'll gradually find/build a support network once covid restrictions ease and once you're able to access childcare for dc. I've also been a nanny and childminder
Definitely contact potentially helpful agencies/people
Women's aid
Health visitor
Social services
Gingerbread
You need support, I'm sorry your mum is unwilling, I know the mn cliché is "grandparents owe you nothing" but personally I think it's weird and selfish of grandparents not to help out and want to spend time with their own grandchildren, unless there are serious disputes within the family - from your post at 1031 it sounds as if she is toxic too. You may find the "stately homes" thread on here and also
https://outofthefog.website
For help and advice on dealing with her.
Regarding getting space/peace to do certain things while dc are awake, do you have a play pen or something you can use as such? If you don't, I would say getting one is a good idea, check out your local free cycle type Facebook pages, freecycle and gumtree. My local Facebook pages for freecycling is ALWAYS advertising things like this as while they're useful they aren't something you use for long and can take up a lot of space, so once their dc are done with them people are pretty keen to get rid!
I used to use dds travel cot in this way for when I needed her safely out of the way and occupied and happy while I did certain tasks I couldn't do while she slept. Eg hair drying, hoovering, certain diy tasks (she seemed to be a flipping magnet for tiny nails and screws!)
Not for long periods you understand but for short periods no more than an hour at most to be able to get certain things done.
She loved it as long as she had plenty of toys to play with and I'd also play music she liked too (nursery rhymes yes but I also went to the lengths of recording the theme tunes of her favourite shows off the telly! This was nearly 20 years ago so no music streaming then much easier to make eg a Spotify playlist for something like this now. The downside is I can still remember the tweenies, hoobs and other theme tunes all too well! 
) so she'd sort of be seated bopping along to the music and playing while I got on with whatever
Maybe tell us what your/her routine is just now and we might be able to make suggestions to make your life a little easier?
And please remember this will not last forever, even within 6 months things will be very different
Again - unpopular opinion - but crying for a short amount of time won't kill him! He'll pick up on your emotion too so if you're calm and upbeat he'll likely be fine. Maybe start off with him in your arms and you're dancing and singing to a song he likes and part way through calmly and joyfully pop him in the play pen set up and just carry on singing keep it all upbeat and fun, make sure he has favourite toys in there
It may take several goes for him to like it/get used to it so persevere - not with him distressed obviously remove him if that's the case, but if he's just a bit grumbly and protesting well it's not going to hurt him to start to learn sometimes he needs to do things he doesn't like - and if it enables you to cope better then that is better for him too
My life is over until DS is 18
I can assure you this is not true!
I went back to uni when dd was almost 4, I made some amazing friends at uni, dd was at nursery and had a whale of a time there, I also made a wonderful friend with one of her nursery friends mums, who was also a single mum. No covid then so yea that's an issue at the moment but won't always be. It was at this point I was able to socialise more and have a bit more freedom - it's a gradual thing. They don't become independent overnight but neither are you completely tied to them until they are.
When they get to high school age there's a lot more they and you can do without each other and by the time they're about 14/15 they're almost totally independent.
This is definitely the hardest stage.
You sound also as if you are quite a young mum? That can mean you're unaware of how life/motherhood works and don't know the ways to make things easier for yourself.
DS has gone for a nap now and I’ve got housework to do.
I wonder if you're perhaps also trying to maintain unrealistically high standards with housework? If so, lower your standards to something basic and manageable.
I have dx ocd and this is a mistake I made and knackered myself (literally)
Do the absolute basics it's not going to be the end of the world.
It sounds as if you entered true adulthood (finishing uni) and motherhood at around the same time?
The more you post the more obvious it is you are depressed, you're unable to see the tunnel let alone the light at the end of it! All you see are the bricks around the entrance blocking seemingly everything. Hardly surprising with all you've been through but also gives you an unrealistic perspective on everything
Be completely honest with the counsellor and has medication been discussed/prescribed?
If your sister is 16 can she not watch him on occasion? I'd think it's the least she can do as auntie for you. Have you asked her?
The last thing I would do as a struggling single mum fresh out of a DV relationship is contact social services.
Why? Is this based on actual experience or on myths?
I've had ss support at points due to my mh issues. They've always been kind, practical and understanding. They've provided or helped me find practical support, given my suggestions and ideas to cope, and even advocated for me to get support sooner from other agencies. They are not the child snatchers some claim! Usually based on zero actual experience with them!
because that life is just over now
It's really not over! It's just on hold for the moment - for everyone! My 20 year old dd is also missing getting dressed up and having a good night out but those times will come around again for both of you.
For me it was when I was back at uni and my other single parent mum and I shared babysitting duties, and I was in my 30's at that point. I'm 48 now and I only stopped going clubbing around 5 years ago and that was because of physical health issues (I'm disabled and I simply can't cope with standing for hours in a busy club if I can't get a seat now)
You also don't have to "dress like shit" now, you may need to dress differently but maybe try posting on the style and beauty board for ideas? Feeling good about how we look is of benefit to us but there are loads of ways this can be achieved and this being MUMSnet we absolutely understand how hard it is making the transition style wise
The playing with him - have you learned anything about child development ever? It can help if you understand the stage he's at and ways to make play enjoyable for both of you
At this age he's only just starting to play with as opposed to alongside and won't really enjoy that until he's older.
What play does he enjoy most? What do you enjoy most? What do you enjoy together? Do a mixture of them and also understand it's not hurting him if you leave him to play alone if he's content/happy and get on with something you enjoy.
It's a case of finding a balance and that takes time and experience to find
It does get easier and you will get more practiced as a mother, nobody knows exactly how to straight away! And nobody's life is perfect not even those families you think of as Disney families