Thanks! Your observations and suggestions were helpful!
It sounds like he is bored, and may feel a little jealous of the new baby. It's difficult to keep a 4 year old quiet, and not run in the house. I think if dad wants this then he needs to make sure he does a lot with his son to burn off some of that energy, and spend some quality time with him. Make him part of caring for the baby, play games with him, keep him in the loop. At 4 they want someone to play with, so it's up to mom and dad to play with him, I'm sure he gets lonely and bored. Otherwise he sounds like a normal child, most kids don't like to make eye contact when they're being reprimanded, and this is why you get down to his level and ask him to please look at you. Tell dad your son is a little guy with a lot of energy, and you both need to find ways to entertain him and make him feel as special as the baby. Babies are very demanding of our time, and he may feel a little slighted by this. He's fine.
I have a 3 1/2 year old whose only life change recently has been going to preschool after a summer break. Yet, some of the things you describe are also true of him, especially not being able to play well on his own for long. I really don't think most toddlers are willing to just play alone, when Mommy is nearby. Maybe the more-whining and throwing things are reactions to change, but don't worry, not everything you list is going to be from the changes.
The way to fight change turning everything upside down, is by developing more routines. At a given time every evening, do such and such. After Daddy gets home from work and has a few minutes to decompress and change clothes, etc., it is Daddy's job to take child outside and run him up and down for a half hour (especially if he wants his son not to be so antsy in the house). Grandma cooks lunch with toddler on Saturday. Mommy gets a long bath on Saturday morning. Etc. etc. etc. If you do these things each time, the routine relaxes the child.
We use post-it notes (the smallest size) in several colors to indicate what day things happen. We stuck them along the baseboard in a row like a train with a lot of cars, one representing each day, and school days are a certain color and Sunday (when we go to breakfast out) is another color and so on. Birthdays get a sticker, as do upcoming holidays. Our little post-it calendar goes all the way down the hall and has reached Christmas (marked by candy canes). When we go to bed at night, we look at the color of the next day and talk about what it means, and take down the just-passed day's post-it and put it onto a standard (gridded) calendar. It helps my son get his head wrapped around what is upcoming, (and if he ever gets interested enough to speculate about the standard calendar full of gone-by days, he will see that they line up too.)
I can't tell you the number of times he has asked "When is Christmas?" or "When is Halloween?" and we've gone to look at the post-its. Maybe a calendar of one kind of the other, so your son can see the repetitive routines of the days, will be helpful.
Otherwise, as always with kids who are dealing with emotional things, give him extra love and snuggles. My husband likes the exercise thing, and I notice it also ups the later hugs-and-snuggles quotient between him and my son.