US-60 from I-55 to Springfield is fine, assuming that it is open (subject to flooding in Spring). It is mostly four lane divided, slowing down for built-up areas.
US-63 goes through some pretty rugged country, a lot of short steep grades and tightly winding road, many small communities with slow local traffic. I got dumped onto 63 once when US-60 was closed by flooding. Luckily, I was driving my wife's Accord that trip, not the motorhome. Average speed was probably 35-40 mph that day.
I don't know SR-13, or at least don't remember it, as it is not one of my regular Missouri routes. It is an important connector, might be one of the better state highways in Missouri, might not. I know I've never been on that section of SR-7. My experience with the state highway system in Missouri leads me to avoid it as much as possible when driving my motorhome. Lane widths down to 9-10 feet are common, some of the roads tend to be high crowned, many do not have usable shoulders and are often in poor condition, particularly breaking away at the edges. FWIW, I have to say the same for state highways in Oklahoma.
US numbered highways in Missouri are better generally, though not always as good as they are in Kansas. I would take I-55 on up to US-60, the higher travel speeds should make up for the longer distance. US-60 to Springfield, then either SR-96 (it is old Route 66) over to Carthage or US-160 to pick up I-49 at Lamar. My problem choosing that I know only shorter sections of SR-96, and have not used this section of US-160 at all. But on the US-numbered highways in this part of the country you can at least expect a 12-foot lane width most of the time, but not always a usable shoulder.