A good rule of thumb is about 1.5X it's DLS (diagonal leg span) to the top of the enclosure from the substrate. Any more than that and a fall from the lid or sides can result in injury. Unfortunately an exoterra cube makes a poor enclosure for nearly all terrestrials (because of those doors) without some mods/tweaks which are too involved to make it worth it, IMO, especially of you have something else handy. You'd best stick with a KK or something similar, where you don't have anything preventing you from piling the dirt on.

Firstly, you'll want to refrain from bothering it too much. They take time to "adjust" to their new surroundings, and frequent removal of the hide or mucking about the enclosure will essentially prolong their period, and result in a stressed, unhappy spider.

Second, forget the sprayer, and don't worry about the humidity being too low. This species is very resilient, and will do fine with just a water dish (just make sure it's completely topped off) even if you live in the desert. If you are worried, you may block the top completely with Saran Wrap with no issues. The water dish will provide all the hydration it needs, and she won't suffocate. If you are still worried, chuck the hydrometer. Problem solved!
Same goes with temperature. Forget what those care sheets tell you. Whatever temperatures you are comfortable with will be just fine for your tarantula. In fact, I find subjecting species to natural seasonal temps to be beneficial for predicting molts, which helps immensely for breeding.

As for cleaning, all you really need to do is spot clean. Boluses, clumps of poo you can see, and dead feeders should be removed as you find them. You shouldn't clean out the entire enclosure at all if possible, only doing so in dire situations (overrun with pests, fungi, or whatever else), if it gets really dirty (webs aren't dirty!) or if you are rehousing. I know some hobbiests with tarantulas on 7 year old + dirt!

Hope this helps.