Everyone should pick up after themselves. We learn that in Kindergarten. If you were working fulltime outside the home, you would still have to do all this work at home. Well, taking care of children, esp babies, is a fulltime job. Therefore, hubby needs to pitch in. When is he being a father? Did his Dad sit in front of the TV or video game every day? And his grandfather? Was his mom OK with it?
Have a talk with him, let him know that you would like to see him and the children clearing the dishes together. The two older kids should be doing dishes by now- with you and your husband's supervision- one of you can hold or tend to the baby during this. Then, when dishes are done, homework or reading for the older ones, reading books to the younger 2; with no TV on until 8 or 9:00, and even that should only be a couple nights a week. Bathing, brushing teeth, PJS on, picking up bedrooms, saying prayers, can all be supervised by him and does not take long.
Let him know how you feel about his dirty clothes, trash, etc. Tell him nicely, gently and that you are dead serious.
You can ask him how he would feel about setting up a schedule of chores. Or maybe if he would agree to Sat AM house cleanup where all of you pitch in to vacuum, dust, clean bathrooms.
I think that women tend to let men step on them. We are not slaves. It is rude for one spouse to sit while the other is working. You should both be working until you can both sit down.
If he sits down and the jobs have not been done, ask nicely "sweetie, could you please help get these dishes done". Men have a tendency to be lazy and forgetful. They want you to say what you want, so don't be afraid. Video games has ruined a lot of relationships.
If he is getting the night off to play, then maybe you should too. If you like to read or play video games, do it. Show him what will happen if you don't work together to get the work done. You have to be somewhat of an actress to pull it off. And when bedtime comes, well maybe you won't be so attentive to his needs if he is not attentive to yours.