My son and your son are about the same age (mine is 6 1/2--he'll be 7 in February), and with school, Choi Kwong Do, and Cub Scouts, he goes to bed pretty exhausted every night. There is no time for naptime during the school year and he doesn't seem to want a nap. But every so often if he's been up extremely late the night before (like, at a sleepover) or he's obviously exhausted, I'll send him to his room and tell him he has to lay down for half an hour and then I'll come check on him. If he's awake after 30 minutes, he can get up, but I can tell you that it never happens that way. Generally after 10 minutes he's asleep. However that doesn't even happen often--maybe once every few months.
If it were me, I'd drop the nap altogether. If he's in a room destroying stuff, he's obviously not sleeping or even on the path to sleeping. I can understand a necessary "quiet" time during the day, but since you all ready know that leaving him in a room unattended is only going to result in destruction, I would move him out of his room, into a public place, and encourage activities that work the brain. He could quietly play with Legos (which are pretty tough to destroy), Lincoln logs (my son LOVES his!), work puzzles, read, a destructo-bin (my son loves to take stuff apart just to see how it works. So make a bin that has stuff in it that is ok for him to take apart. Old electronics, for example. Trust me, a 7-year-old can handle a screwdriver).
Also, and this might not be a popular sentiment, but if he's not involved in some kind of physical activity every day, he needs to be. I'm not saying you need to sign him up for soccer or football or anything at all like that, but it really sounds like he could use the opportunity to really work off some energy. All kids are big balls of energy that needs to be expended in some physical way. I was reading an article a few weeks ago (which, God help me, I can't find now) that suggests that children's body and spatial awareness have declined significantly over the past 30 years or so because children simply aren't spending time outdoors. I want to say that the study recommended that children have something like 5 hours of unrestrained outdoor play a day--a difficult task for my family to fulfill, but 30 or 40 years ago would have been entirely do-able. Anyway, so children are less coordinated and less aware of their bodies in space. My point to this is that, besides wearing him out, getting him physically active has other benefits.
I've found, recently, that punishing my son for stuff is counterproductive and winds up with both of us being miserable, and I've just accepted that, for me, I'd rather encourage positive behavior when I can. Your son is 7. He knows that tearing up mattresses and pillows or books is unacceptable, but he does it anyway. I would sit down and talk to him about it. The conversation might go something like this:
Dad: So, we need to talk about your destroying things when mommy and I aren't around. You know that's naughty--right?
Son: Yeah (or maybe he'll say no. But let's pretend that he says yes.)
Dad: And you know it makes mom and I really sad when we see that you've torn up a book that we know you really like or drawn on the walls of your room (or insert whatever most egregious thing he's done most recently might be).
Dad: And you understand that it's important that you have some kind of quiet time during the day. So sister can nap and mommy can empty the dishwasher and refill it again--right?
Dad: You're a big boy and mommy and I need to trust that we can leave you alone in a room for a little while and you won't destroy it while we're away. That's a big responsibility, but we know that you can do it. So instead of taking things away when you destroy them, we're going to make a chart and every time you are left alone for a little while without destroying stuff you get a sticker. And when you get 10 stickers, we'll do something special (insert whatever special thing you think he would like or have him help you decide what it is).
And let the conversation go from there. He might say No, he doesn't understand that it's naughty and that's the window to have a more in-depth discussion with him about it. If he does understand that it's naughty, then you can move forward from there to correct the issue. You could ask him why he does it, but I'm going to ask that his answer will be "I don't know" which I've decided is kid lingo for "I was bored and just wanted something to do". Although, he could say "The zombies in my closet told me to do it."
In the meantime, you do have to provide him with something to do during that quiet time that he's unsupervised until he can be trusted to be left alone with nothing to do and make a good decision about what to do with his time.