You can tap and hold on a route and select 'Download'. Just opening the route once will also work. As long as you don't change the route after that, it will be saved on your device. Just make sure you have also downloaded offline maps, otherwise the route can't be calculated if you don't have an internet connection available.
Routes downloaded as described in message #2 are not available outside the app.
If you want to download the route to use elsewhere, for example as a .gpx or .itn file, you can sign in to myrouteapp.com with your account, open the route and use the 'Save as' menu to download it.