It seems my filters and saved searches from the Web GUI do not carry over to the console application? Is that be design. Or is there some checkbox I'm missing that syncs those two?
That is correct - you can export/import your settings, though. The AIR Console settings are stored in your user profile while the web console settings are stored on the appliance.
From the Console, in Manage > Appliances, hit (top right) Gear > Export User Settings to export, then Import User Settings to import. This import does overwrite everything, it's not a merge operation.
Very nice. Thank you Nicole.