We've done a few road trips with toddlers and they were always well behaved and usually slept most of the time. We usually stop every couple of hours to stretch our legs etc. Take her favourite toys and lots of snacks.

The motel won't care where Jazz sleeps. Most motels have portacots they can put in your room if you request one.

I don't think she'll get homesick or unsettled as long as you are with her. She will probably enjoy the change of scenery as much as you will.