generally yes. Although custom UI has not been implemented yet for device handlers in New so some device handlers may have limitations. You install them the same way, regardless.
Yes, but dont use both SHM and STHM at the same time and dont use device presence in both Classic and New at the same time.
(www.rboyapps.com - Making SmartThings Easy!)
To add to what Nathan said, see the image on the first post which explains how ST works, you’ll see that the new/classic apps are just different user interfaces to access your account on the platform. So the core it the same, how they’re displayed are different and then each app has access to certain features on the platform that the other app doesn’t (e.g. new app provides Automations, the classic app provides custom UI’s etc).