I read that pilots often use printed checklists.
Is this a legal requirement, or are there also electronic versions available?
Yes they do, and according to the FAA they're better than paper ones. All sorts of aircraft can have electronic checklists, from large airliners to small Cessnas: modern avionics like the G1000 support checklists and the older but very common GNS430 does too.
In the US at least there's no legal requirement to use checklists in any form - electronic or paper - with some caveats:
As for mobile devices, there are already many EFB applications and standalone apps that support checklists. In some cases the app provides a template for a specific aircraft type, in others you have to enter it all yourself. There may be some copyright considerations, but third-party checklists are very widely available so obviously it isn't a huge issue.
Finally, it's worth noting that many pilots - especially those who own their own aircraft - customize checklists for their own needs.
Newer, more advanced, more expensive aircraft ever more come with computers that have checklists built in that can be displayed on a screen on the flight deck. Think the Boeing 777 or Airbus A380 for example.
This might be optional or standard depending on aircraft type.
Are there regulations preventing such things from being turned into software for mobile devices? I doubt it, long as the device itself is approved and the software can get approved for reliability and accuracy.
But you'd be unlikely to be able to make something like that unless you're working for or contracted by an aircraft manufacturer as you'd not have access to the data needed for the software to display. That data isn't static and is copyrighted (so you can't just take an old paper checklist and turn it into some html pages for example).
You'd have to have some way to update the checklists as new versions are released, and that'd mean having channels with the manufacturer to gain access to their data (which may not be available at all electronically, you'd likely not be able to get it except for the same aircraft that already have such systems built in, making the entire idea superfluous).