Generate .apk file from Eclipse

Generating an .APK in eclipse is possible through

Android Tools > Export Signed Application Package or Export _Unsigned Application Package (if it does not require your .apk signed with the Keystore).

To upload to the Play Store it is necessary to sign the application with your Keystore so I suggest using the Export Signed Application Package option, this will require the Keystore and its password.

Yes, it is possible. You must right click on your project:

Android > Android Tools > Export Signed Application Package

You will need to generate a KeyStore that will ask you during the process, it is simple but the generated file saves it because when generating a new version to be able to update in Play Store you will need to export with that KeyStore.