Hi, we have no supported way of doing what you want (will write down as a feature request though) at the moment.

The folder you found is part of our installers so no, you should not copy/move it as it may or may not screw up any future updates on your system.

I really would not recommend copying files across different computers but if you really really want to try there are a couple of things you should know:
1) First of all, as you suspected there is a file that stores the favorite state, this file is located C:/Users/”Your user”/AppData/Roaming/Toontrack/EZdrummer/MidiDB
2) This file, while storing the favorite state is actually storing ALL data regarding your MIDI packs on that system.
3) Any user MIDI (added folders etc) are stored by absolute path so they will not work when moving this file.
4) If your other system is missing a MIDI pack it will still show up in your browser however it will show the files as missing.
5) There might be other issues as well as the MidiDB file is intended to work only on your current system, I can give no guarantees that it will not cause crashes or cause other bugs.

So, my recommended approach (well I don’t recommend doing this at all and take no responsibility if something fails horribly!)
1) Make sure all EZdrummer instances are closed on both machines.
2) On your laptop, make a copy of the MidiDB and MidiDB_fav file (if there is a MidiDB-shm or MidiDB-wal, copy those as well) and store somewhere for backup!
3) Remove the MidiDB (and -shm, -wal if present) from your laptops system as well as MidiDB_fav
4) There is a folder “OnlineMidi”, make a backup of it and then remove the folder from your laptops system (or just rename it to OnlineMIDI_backup or something)
4) Copy over the MidiDB file from your desktop to your laptop (again, if -shm and -wal files are there, copy them over as well, do not copy the MidiDB_fav)
5) You should now be able to launch EZdrummer on your laptop (fingers crossed)

Hope it helps