He sounds to me as if he may be bored at preschool, as well as not supported particularly well there. If they are handling discipline differently to the way you do, it is very hard for him to sort out how to be. I do sympathise. I had two very active, vey bright, very easily bored boys too. They are hard work at this age.
If it were me I would a) ask if we can cut down to three days a week and work on behaviour, then build up to 5 when he is settled. That will give you a break 3 days a week and you might find you enjoy your 2 days together more too.
I would ask the preschool to be more structured in their approach. Instead of saying "Please don't . . . can they suggest a positive alternative to him and distract him too" eg please don't tip up the toys, it makes a mess. Let's put them away (you win a sticker if you do) and then we could go on the trainset" Buy a notebook and a set of stickers for them and ask them to give him a sticker if he does something good. Then you can talk about the good things later on. You can also set him targets - no kicking people, no locking the loo door etc and you can give him a sticker if he doesn't do it too.
I would ask if I could go in and observe him, or even work along side him occasionally, though it might be stressful for you. You shouldn't be naughty stepping him a lot at preschool, as he needs to adapt to their system, but you could agree on a strategy with the preschool, for how much involvement you could have. It also means that you can back them up by talking to him about what he does at pre-school, or by giving him extra rewards (or punishments) for appropriate/inappropriate behaviour. This worked brilliantly with my ds1. (Warning - it was a disaster with my ds2 who was always worse if I was around!)
If you do have him at home for a couple of days, have a range of structured activities to do at home - 30 minutes painting, 15 minutes tidy up, 45 minutes walk to the swings, 15 minutes hang out the washing together, 10 minutes snack time, 30 minutes lego, 5 minutes tidy-up-race, etc etc. You don't have to stick to the routine, but have it planned out so that you don't sit around wondering what to do . . .
There is also swimming, biscuit baking, playdough, feeding ducks, supermarket shopping (mine loved that, once they understood that they could make a list before we went, with pictures of items they needed to buy - apples, squash, weetabix etc. Making the list takes ages, then they have to find the items at the supermarket and put them in the trolly . . . .
Hope that helps a bit. Best of luck