Mine didn't speak until 16 months old. He was busy doing other things. In his case, he was an early walker (8 months). Yours is growing in other areas - you said he has a good play sense, imitates well, and so on. He's probably not doing things he did at 10 months because a) he's already mastered those skills, even if he doesn't use them all the time and b) he's focused on what you are trying to teach him to do, like say certain words.
It's possibly he is speech-delayed, but it's just as possible that he's developing other skills and will get to the speech later on. I found, with my son, that I anticipated his every need so well that he really didn't have to speak anyway. The doctor said that usually kids with significant problems are delayed in more than one area, but that she would have him evaluated at 18 months if he still wasn't talking. At 16 months, he was pointing and sort of grunting, and I got sick of it - I said with frustration, "Whatever you want in this world, you can have, if you just stop that annoying noise." Lo and behold, he announced "Cookie" - which was in the cabinet he was pointing to. He never stopped talking after that, and moved quickly into 2-word and 3-word sequences. He's now extremely articulate.
I guess I would suggest to try to stop TEACHING him and just let him grow up and develop in his own sequence. Don't anticipate his every move, don't ask him to give you too many things or do things on your command. I think it's good to start teaching skills like "clean up" but really scale it to his age. Let him model your behavior by showing him how you put 1 toy in the toy bin and see if he can pick up another toy and do it. It's not so much to get him to "perform" but to begin early with modeling desirable behaviors, like tidying up. But again, keep it very age-appropriate.
It's really not unusual for a 13 month old not to speak or walk. He'll get there. I would try to expose him to as many activities as possible - get him outside in nature, and let him play in the dirt and pick up a rock. Take him to a playground and let him go in the baby swing and start to climb on some of the equipment. Take him to a children's museum and let him explore textures and shapes. Go to the library and get out some children's books - sometimes they have story hour or puppet shows, and our library has a puppet area for free play. Be sure he has toys that require him to DO something - blocks, wood puzzles, touch-and-feel books or lift-the-flap books, balls that roll and that are a good size for his hands, etc. Those sorting toys are fun - he might be young for "put the square block in the square hole" but try one of those hammering toys that has the child hit a ball with a mallet and push it through the round hole. We had one with multi-colored balls and several holes, and when the ball went through, it rolled down a chute and out the other side -- kids like "cause and effect" and it's a learning skill as well as a motor skill. Go to a pet store and let him look at the turtles and the mice and the hamsters. We have a local garden center with an animal area that is free - just watching him learn which animal is which is fun. I didn't drill my son, but I said, "Where's the turkey? I can't see him? Do you see him?" If he pointed (even if he didn't speak), I said, "Oh yes, there he is! Thanks for showing me!" I made it more exciting and cooperative than "training" or "drilling". Kids just don't want to perform on schedule. They do things in their own time.
If your child needs some early intervention services, you'll know in 6-8 months probably. For now, don't worry and don't try to "fix" anything. Just give him opportunities and let him reach milestones on his own rather than on the schedule in a book, which is really just a list of averages.