The application data error is due to a change in how PureBasic 4.20 stores its registry keys. This has been fixed for the next release.
For the old installer you can modify this registry key to point to your application data folder:
http://puregdk.com/files/upload/pgdk1.0.3-4.20_installpatch.zip
The engine core error is most likely because you are using an incompatible version of DarkBasic Professional. Use only the officially supported version 6.7 with PureGDK.
The next release will support PureBasic 4.30 and DBP 7.1.
Also be sure to install the 1.0.3.1 patch which resolves an bug where PureGDK executables would crash silently at startup on some configurations.
To install it replace PureGDK.dll in your Application DataPureGDKplugins-user folder. This folder is in your user accounts folder under Documents and Settings or the All Users folder, depending on your choice during installation.
http://puregdk.com/files/upload/pgdk1.0.3.1.zip