How do I make sure my laptop stays asleep when the lid is shut?
Be careful what you leave open in the background. Chrome's minions ("helpers") seem to work even with the computer asleep. Some web sites even in Safari can stay active when you think the computer is not. Close everything before putting it in your backpack.
New computer are thankfully not like the old ones we graybeards had to use by candlelight . Old models took a long time to load, wake, do anything, so we tended to leave them in idle mode and accept some heating.
New computers start from cold faster than an old hard-drive system could launch one app. Leaving them off or with all apps closed is not longer a chore.
I have not experienced lid closure issues since my early silver-key Macbook Pro died. They suffered uneven lid closure due to lid bowing, a design defect, and I resorted to rubber bands just like my esteemed colleague D. I. Johnson suggests. For me the Unibody design solved the problem with better lid closure, but a band is insurance. However if software is the issue, the rubber band won't help.
However, check that you lid fully closes for one very good reason: a swelling battery can affect closing.