S.B.
As far as getting things done,my friend gave me a life saving suggestion, try to include my son in the work. I get sooo much more done. With his "help" it can be slow going, but it's a lot more productive than waiting for nap time. When cleaning the bathrooms I have him wipe down the front of the cabinets. He used to be happy with just a paper towel, but lately I have been giving him a baby wipe (he thinks it's the Chlorox wipes mommy uses). He helps with laundry by helping me carry and sort clothes. I bought him a kid sized broom and a vacuum to help sweep and vacuum (the vacuum was bought for $2 at Goodwill and has a sweeper on the bottom so it "really" picks up dirt). I bought him a duster and have him do the blinds while I dust the breakables. He has a step stool in the kitchen that's tall enough for him to help with dinner and dishes. Sometimes, if I can't include him in dinner prep, I will give him an egg to beat...that has been known to occupy him for a long time! :)
To stimulate learning and thinking I just try to include it in our everyday activities. We read A LOT! We count and talk about colors with many of our chores. Sing the AbC's. Talk about the letter sounds we hear. We go to Spouts and talk about the color of the fruits and veggies and then we weigh them to work on number recognition. I try to get him to problem solve on a toddler level, like today the trash was full and his wrapper kept falling on the floor - I had him figure out what we should do about it. And now that he's three, we have started talking about time . He doesn't get it, but I figure it doesn't hurt and it reinforces number recognition.
As far as how much time he spends alone to play, I had to sort of "teach" him that skill and we still have days where he wants mama the whole time. It took a while, but he will spend a few hours playing alone - I reorganized him room and made everything more accessible to him and that helped.