When does Larry sleep? He's never shown having just woken up or groggy or anything.
Well, the older guards in the first museum did say that the tablet gave them new vibrancy and energy. Maybe, since Larry's younger, it's just giving him health and energy to avoid the problems of sleep deprivation or whatever.
One might get annoyed at how the real-life historical figures are Flanderized to their most (in)famous trait, but that's often how real-life museums portray them. The wax mannequins represent the museums' view of history.
A few of them actually complain about this.
Also, the first movie implies that the exhibits absorb knowledge and information from around where they are, such as how the mummy can speak English because of his time in Cambridge. The Roosevelt statue in particular essentially admits he's just playing a part, presumably based on what he's gleaned from guides' and visitors' conversations over the years.
Which, if it's true, would be bad news for Rebecca, who isn't going to get anything useful out of Sacajawea for her dissertation.
In the Italian adaptation of the second movie is implied that the exhibits are somehow fully aware of critical pieces of information about modern times: Napoleon is able to keep tabs on one of his potential modern descendants and relate about him to Larry even if he simply couldn't have had no way to know about their relationship.
Amelia Earhart claims to have always been aware of her fictional nature, and the bust-Roosevelt shows no surprise about the existance of a wax-Roosevelt with arms, and legs and an horse. Thus, it can be supposed that the same magic keeping them alive gives them awareness of their nature and roles.
Jedediah is able to drive the remote controlled car from inside. How is this possible? The tablet brought it to life too! His gun may not have worked, but the triggering mechanism did; we heard it click when he tried to shoot Larry. Likewise whatever they used to set off the ineffectual dynamite. And the car has a working motor inside; all that's needed are working controls.
I realized the other day why an ancient Egyptian artifact might reanimate both the dead and lifeless statues. According to their mythology, a person's sheut or "shadow or image" is just as much an essential part of a person as their soul. So reanimating an image of Teddy Roosevelt would make as much sense to the ancient Egyptians as it would to reanimate their body: both are equally important and both are considered part of that person.
In the sequel all paintings and presumably pictures are brought to life as well as the exhibits. Fine and dandy if you're kissing a girl or sailing in stormy seas, but what about the Holocaust exhibits?
Remember how Ahkmenrah spent most of the first movie locked inside a stone coffin, screaming his head off? How long has he been in there? And on that note, how many centuries did he spend coming back to life in his own tomb, knowing that he'd be dead again come sunrise?