Everything will be feeding into other each other here - if you hadn't had dinner for several days you would probably also be screaming and shouting at 3am too - so to try and break it down can we help with some ideas about eating firstly?
For example, when you say he doesn't eat, do you mean he would not eat anything at all? My dc lost weight very quickly when young and I found I had to be very inventive to get nutrition into them - for example one would only eat porridge, spaghetti bolognese and cake, so I would put egg and honey in the porridge (which actually tasted pretty good a bit like custard) to get some protein into them, I would put courgettes and carrots in the bolognese so it had more of the 5 a day, and I invented a cake which had eggs, honey instead of sugar, milk and ground almonds. It was trial and error, I remember doing about 5 dishes in one sitting once. I agree with pika here, the advice that they won't let themselves starve just does not work with some kids. A well fed child is going to be happier and have more energy and sleep better, and eating then gets easier as time goes on.
After getting some food into them, get out for some serious fresh air and walking. If it is hard to get them to walk, take their hand and point out birds, houses, etc so that they are distracted. A few rounds of the local play area to exhaust them and build up muscles. Do you have furry snowboots and very warm coats and hats for them? That will make them feel snuggly which helps.
Better food and more exercise will help with the mood and with the sleeping.
With the TV I would try to go cold turkey and do things he would accept as alternatives. For example at that age me playing with dc and entertaining dc, or reading to them, or doing painting or play dough or something with them would be acceptable alternatives. Would that work? It may take a bit of practice, but the more time you spend doing these sorts of things, the more they will want your company instead of TV.
Are you all clued up on what activities and toys are best to do with 3 year olds, the "right" stimulation for his age? I found getting it right was like a switch going on or off in terms of behaviour, a sudden "aha" moment. Does he have some stimulating toys and activities for at home? Thinking 6 months ahead is also good, in terms of activities, so you know where you are at.
It is really hard work, but it is worth it.
It does sound to me as though you are struggling and need some help in real life too, though, your HV is aware that there is a problem with eating which is good. These posts may all help though.