Do pilots use electronic checklists?
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:


They're considered a fundamental best practice in aviation, so if you don't use them then someone could argue that you're acting carelessly or recklessly
Incorrect - especially missing or incomplete - checklist use has been linked to many accidents (see this question too) so clearly it's in your own best interests to use them properly
If you're flying for an airline then your OpSpec probably requires use of checklists, in which case they're mandatory


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.