In my opinion, I don't think that it is necessary to teach him how to read yet. If you have that much time before he goes to school, you can wait on reading and start focusing on his listening and socialization skills. Many of the things you mentioned he should be doing by himself. The best way to get him to do it, is by making a game out of it.
When you want him to shut the door, sing, "One, two, three, four, please shut the front door." Sing it three or four times in a row. If he still doesn't do it, I would suggest having a consequence. I don't mean spanking. I am talking about taking away a special toy when he doesn't listen, timeout, etc. I am lost as to how to get him to clean up, as I am still working on that with my two kids, who are four and five. When I need them to help clean-up we usually make a game out of it, shooting "baskets" into the buckets, toyboxes, etc. I also say, "Who can clean up the fastest, whoever can, gets a treat(piece of gum, fruit snacks, etc.)" Today when I was trying to get my four year old to clean up, I got out the vacuum, turned it on and said, "You better hurry before I vacuum everything up!" She squealed with delight and picked up every single last toy, I then chased her around the living room pretending to "get" her with the vacuum.
I guess the advice I have is to have fun with your son, make cleaning fun and give him more responsibilities. Make a chart on the computer and give him a sticker everytime he brushes his teeth and gets ready for bed by himself. My kids get their own cereal in the morning and get dressed by themselves everyday. They don't always do it in a timely fashion, but they get it done. We put a shoe rack in our hallway with the kids shoes, so they always know where they are. If there are no shoes on the rack, I send them on a hunt for their shoes. You can also keep a basket by the front door, but I found that the rack was easier, as we can see what shoes are on there.
As for the learning aspect with him, just play games. You can print off free flashcards online, go to learning websites and let him play on the computer(my daughter likes Nick Jr and Funbrain) Count dry beans, hide objects in rice or flour, put word labels on objects around your house, etc. I could go on and on. Just google learning activities for preschoolers and you will find a wealth of information. I hope I have been of some help. Parenting is not easy, it is all about trial and error. I make mistakes daily, but the important thing is to learn from your mistakes:)