Most kids have trouble with a lot of this, and children who are in school with a whole room full of kids with sequential tasks often struggle.
When you say you "send him back" for socks or underwear, what does that mean? He's showing up in the kitchen for breakfast and you are sending him back to his room for (in his mind), one thing? For many kids (most, maybe), "get dressed" is too vague. So is "clean up your room" and "get ready to go."
So, the night before, set out tomorrow's clothes, giving him some choice in the matter but not totally up to him. Base it on the weather and on the day's planned activities (Grandma's birthday vs. school field day, etc.). Limit choices if you have to: "Do you want to wear your orange shirt or your yellow one?" (Show him both shirts and let him pick between the two - don't just use the verbal "orange" or "yellow" labels.) Repeat with pants, socks, underwear. Place the clothes on a chair or other surface with the first things on top (underwear) and then the next things, followed by socks. You can talk him through this as you do it, but put them in order. Socks sit on top of the shoes he will wear. If there's no laundry basket or hamper for dirty pajamas and yesterday's dirty clothes, add one to a corner of his room or the closet.
Put a hook by the door for his backpack. It always goes on the hook. (In school it always goes in his cubby, I imagine.) But there's always a place for it. If you have to, put up a double hook and put a sketch or internet photo or actual photo of his backpack over one hook and a photo of his jacket or coat over the other hook.
Remember that some kids develop visual skills before auditory ones, and other kids are the opposite. So just giving verbal instructions doesn't accomplish the goal. While it's clear to you because you have good verbal skills and see the end result in your mind, he might not be that way. Maybe he's not strong with auditory processing (doesn't mean there's a problem, may just mean he's not there yet), but is better with visual cues.
Do the same thing with his lunch box - put up a photo on the fridge or cabinet showing an open lunch box with the components in place: sandwich in a baggie or square plastic container, juice box/water bottle in its proper corner, ice pack in its place, snack bag or dessert in its proper location. Then show him how to line up his actual lunchbox with the photo and match them.
Do you say, "Clean up the playroom"? Or do you have marked bins for different categories of toys (hot wheels, legos, balls, etc.)? If not, get storage bins or one of those plastic drawer units, and label them (words and photos) with "crayons" and "markers" and "paper" and "paints" and "cars" and whatever else. Then practice focused and limited clean-up: "Pick up all the crayons and put them in the crayon drawer." Only let him pick up crayons until they are all done - that lets him practice staying focused on "crayons" only. Then "pick up cars" or "I'll pick up cars and you do the balls." It's so much more specific than "clean up."
Find other ways to play games that match things and test his memory. Do those "one of these things is not like the others" puzzles; do a jigsaw puzzle with the box cover nearby (compare/contrast), play the "memory tray game" (put out a tray of household objects - a key, a pencil, a toothbrush, a quarter, etc., cover with a towel; remove towel and study for 30-60 seconds, replace towel or remove tray, then see how many things you can remember). Start with fewer objects and less time, build over time. It's a good way to practice memory skills as well as writing skills, but if he's still an emerging writer, play it cooperatively with him listing and you writing them down. Next turn, have him make up the tray and you play the role of guesser.
Sometimes it's not a matter of a diagnosis of ADD or other focusing issue; sometimes it's not an auditory processing issue. Sometimes it's just a kid who hasn't been required to practice these things. So break it down into manageable steps. Yes, you know how to "clean the kitchen" because you've been doing it for years. But he needs to learn to "put the silverware in the dishwasher" and skip the plates and pots/pans. Know what I mean?
If he's not getting nagged at school and home both, he will improve. But as school is out for the summer, you might want to talk to the teachers about things to reinforce all summer. Maybe it's just "put your name on the top of the paper" even if it's just a sketch that's going on the fridge or being send to Aunt Tillie. ONLY work on "name on paper" and nothing else for a week. And do a paper of some sort every day. Week 2: add another skill. 30 seconds, but a new skill that's practiced every day.
You'll see a big improvement if you are very structured and consistent.