Can I read Adobe-Digital-Edition books on my Android device?

Yes, You need to install compatible app. 

Look at Bluefire and Aldiko for general use (including library books) and Overdrive app for library books.

There are also apps from some of the major booksellers.

For DRM books you need to make sure you register with the same AdobeID you use on the PC.

(For non DRM books, don't bother with ADE on any PC, use superior free Calibre instead, or download direct to device.)

You will need to copy any books using a file manager, not from ADE.