You already have a lot of wonderful advice here... We are just entering our second HS year with a 6 year old with many of the issues you mentioned, a 4 year old who has my same learning style, loves to do stuff and sit and learn and a very very busy 2 year old who just likes to explore and break stuff.
I have found with Aiden (6) that then more I try push for hardcore structure the worse it got. I decided to do more unschooling and allowed him to choose what he wants to work on. He currently has 4 daily things he chose to work on (violin, maths, spelling, writing) and 2 other "just for fun things" (genetics and rock strata) and 1 lifeskill (rollerblading - don't ask, I didn't understand his answer! lol)
We mix with a LOT of unschoolers and from what I see of the older kids, they are great with setting their own routine. They will order their own time automatically simply because they are doing the things they really really want to be doing. And I can see as I have let Aiden do this, that in the first 5 months he had epic meltdowns because he forgot to do x or was too busy playing to do y. And I kept saying that it's okay, it's your choice how to spend your day - with a hug and a smile. (Even though inside I was screaming with a little "told ya so")... Now he is starting to realise and self-regulate already. Not every day, not all the time, but he is only 6 and I recon that if he can figure out on his own how to manage his own time to get everything done that needs to be, then he will OWN that for always. We have work periods in our day where I will say, okay Nathan wants to do maths now, which of yours do you want to do? WE work for about 15 minutes at a time and then have at least an hour break for lego, baking, walking, riding bikes and climbing trees, swimming, building puzzles, reading stories whatever. So our day is spread out and feels lazy and relaxing and I get some of my own work done inbetween as well as basic tidying up.
It works well for us.
best of luck as you find your feet!
If he is really having a rough day I help him make a list of what he wants to do that day and he ticks it off as we go (including brushing teeth and watching specific shows)
Screen time - I limit to 40-60 minutes per day only. 50% of that is "learning screen time" and the rest is whatever they want to do on the PC. They have subscriptions to reading eggs, dreambox, a typing program and timez attack and a variety of learning based games. They have city builder series games, plants vs. zombies and wizard 101 for "fun time". Sometimes they google their current favourite show and play those games.
They can watch one mythbusters or "how its made" with DH or the occasional national geographic type documentary. Friday nights is family movie night.
If it's hard to get going in the morning then start the morning slowly - that's the joy of homeschooling. I have come to realise that there is no right or wrong way to do this: all that matters is that we are having fun and laughing together, because that means learning will happen.
