The most "trail friendly" route is probably the Eastern TransAm Express from Washington to Walden, CO. It uses rail trails from Washington, DC to Pittsburgh then the Katy Trail from St. Louis to Windsor, MO. From there it continues briefly on the rail trail then is mostly, but not entirely, on fairly quiet county (country) roads to Walden. ACA has a digital map (app) and I believe GPS data if you want it.
If you want more trail, you can go from Windsor to Kansas City on the Katy though you will have some city riding with bike lanes in KC. I have routes that will connect you from Kansas City to the Flint Hills Nature Trail (rail trail) from Ottawa, KS, to Council Grove, KS, a lovely packed crushed granite trail.
From Council Grove, I have routes I can get you that take you to Rush Center, KS, where you can take ACA's TransAm into Pueblo, CO, on lower traffic count state highways.
The above Eastern Express to KC to Ottawa to Pueblo has the most trails.
As far as storms go, just pay attention to the locals and your favorite weather radar map because the weather service will send out an alert even though the storm is 25 miles away and headed in the wrong direction. Most violent storms are over with by mid-June in west of the Mississippi though they can still pop up. The "tornato alley" runs basically from Lubbock, TX, to Chicago (about a 700km wide swath) though Nebraska can get its fair share also. However, again, most are done by mid-June when it starts to get hot. Honestly, that would not be a big concern of mine at that time of year.
Tailwinds, John