Not allowing sleeping on 24s is dangerous, I would never do it.
Our company policy is that 24s are allowed to sleep on their 0800-0800 shift after 6pm assumming station chores are done (taking out the garbage, sweeping the floor/vaccuming the bunkroom). The main work area, showers and bathrooms are done by a contract custodial service but the custodial service is now allowed in the bunkrooms or the garage (which is cleaned by the mechanic crew). Day shift (those that do the 8 hour, 10 hour, 12 hour, and 16 hour shifts) are generally not allowed to sleep, but there is no written rule that forbids it, its just that you are usually so busy that you can't.
In addition to EMS duties, if we are posted back at station, then the boss will sometimes ask us to help sweep the garage but not much more than that. We are of course required to check out our own trucks. For most crews this takes about 20 minutes, but for me since I'm a contingent and work on different trucks everytime, it can take upwards of 45 minutes to an hour, especially if the previous crew left the truck with items missing or if there are maintainence issues with the vehicle. ( I really wish we would get all the same trucks for our fleet, but alas, that is not cost effective)
The worst days, though, are those that start at 0530, and when we get in we are called out immediately which means I'm now on a truck i'm not 100% familiar with, that might have been last used a week ago by a crew that was running 2 hours late to get back so they didn't do a post run inspection, with another EMT who might also not have been on this truck in a month. I specifically recall one of those days where our morning truck check didn't get completed until 4pm, and even at that point we wern't back at station so what we were missing wasn't imediately replacable. Luckily, my partner that day had worked on that same truck every shift for the last few months so she knew where everything was and had restocked most of the stuff after her return the previous day.