Yes, they can work over cell/mobile/remote connections. Native Hubitat dashboards as well as the integrated SharpTools.io dashboard (which I prefer).
Hubitat provides local and cloud links that you can bookmark or save as a home screen item. So you can save an icon that will work when you are on your home network but your Internet is down (ie: local) as well as one that will work when you are away and not on a VPN back to your home (ie: remote/cloud).
Sharptools.io is an app on your device. It does require an Internet connection
With Hubitat's native Dashboard, you setup your dashboard and then they provide local and cloud links to the Dashboard. If you want local access, use the local link. If you want cloud access, use the cloud link. Both go to the same Dashboard(s).
MakerAPI provides a world of ability. It's a playground for those that like to tinker with programming and using API's though.
If you want remote control of your devices, you can use either the native Dashboards app or the SharpTools.io app. If you want more control, you need a VPN from remote into your network where your hub is located.
Just about anything in SmartThings can be ported to Hubitat.
See this topic https://community.hubitat.com/t/app-and-driver-porting-to-hubitat/812