On the router issue, it sounds like the rest of their devices are hardwired (since they think they don't have wi-fi), so they either have a really old router, or they don't have the router settings enabled correctly for wi-fi devices. Try to find out the model number of thier router, so we can point you to the correct setting or their router.
On the second issue, try totally powering off, then powering back on to clear the tablet's memory. Also, some Nook colors used to have issues like that when owners tried to use them while they were charging their device. I've seen at least one or two Nook Tablet users who also identified issues with either their browser (wild scrolling or bouncing back to a prir page) or ebooks (page jumping) when trying to use while charging.