Found the logs. It is the command
/opt/alien/system/bin/logcat

The errors I get installing a bought app are:

W/ActivityManager( 6952): No content provider found for permission revoke: file:///data/data/com.android.providers.downloads/cache/downloadfile.apk
W/ActivityManager( 6952): No content provider found for permission revoke: file:///data/app-private/vmdl[randomnumber].tmp
W/dalvikvm( 6952): threadid=16: thread exiting with uncaught exception (group 0x40e39360)

Two first errors seem to happen also when the installation is successful with free apps, so they do not mean much alone. However paid apps are most likely signed with a key to prevent piracy, so they might trigger some checks that cause the Gapps to crash.
Yet again you can install the same bought apps downloaded from same origin so it is not a full incompatibility issue with alien dalvik.

Buying procedure in itself is working; I have bought both apps and music through GooglePlay on Sailfish.

The 'No content provider...'-error also seems quite common with Android homebrew ROM, and common cure that is chmodding (o+rwx) to following directories did not help:
/data/local/tmp
/data/data/com.android.providers.downloads/cache