1) From Ashtabula, I think you take I-90 over to Cleveland, I-71 to Cincinnati (actually about 20 miles south of Cincinnati...I-75 and 71 are the same interstate for the first 20 miles of Kentucky), then I-75 to Corbin.
2) Corbin is a fairly large community. No, it's not a Cincinnati. Or even Lexington. But there are a number of hotels and restaurants at the exit. The very first KFC is in Downtown Corbin.
3) You do realize that you'll be on US 25E for a couple of hours, don't you?
4) Hope you realize you'll be going through a mile-long tunnel. That's just outside Middlesboro, about an hour east of Corbin. The tunnel marks the Kentucky-Tennessee border. The tunnel is in the middle of Cumberland Gap National Park.
5) The only mountain you even have to concern yourself with is Clinch Mountain. That's about a half hour the other side of the tunnel (and about a half hour outside Morristown, where you pick up the expressway). Not winding at all. Straight up, then straight down. No problems, especially in the summer. Beautiful drive. At the top, you can stop and view the scenery.
Since you'll be travelling leisurely, let me know if you have any questions regarding the route; especially between Columbus and Morristown. (Cincinnati in particular.) I can tell you where to stop and what to avoid. Good luck.