Hello and welcome to the RTA forums!

Whereas it is impossible to predict the weather this far ahead of time, you can get an idea of averages for the time of year by either looking at a weather site like wunderground or the weather channel, or at the nps.gov site and select your park. Bryce is at high elevation and you can at the least expect it to be cold, and well below freezing at night.

The roads, although mainly 2 laned are all paved and in good shape and under "normal" circumstance shouldn't offer much of a challenge, of course weather can also play a part in how that pans out.

You can expect it to be 2 very long days of travel to get there, which might prove too long with kid's in the car, so 2.5 to 3 days with some stops either way would be better suited I would think. Have you got just the week for your travel plans or longer ?