So should I just bleed the air out of the system or should I flush it out with a hose as described, then bleed it?
I flushed my Passat as well. I did what some say is a bad idea, but I put CLR in the inlet line and let it sit for about 20 minutes. I then ran the hose through the inlet and had a long outlet hose attached and running to a bucket to collect material for inspecting, also, coolant is bad to let run-off.
I got so much gunk and build up. I actually did this because she wasn't getting ANY heat whatsoever and the windshield wouldn't defrost. I ended up flushing my inlet and outlet 4 days in a row and now she gets good heat.
The trickle sound could be a number of things:
1) A clog in the heater core or inlet/outlet line which slows water movement and creates a dripping affect, causing the sound you hear.
FIX: Just flush the lines repetitively, first through the inlet line, then through the outlet line.
2) Air in your lines.
FIX: Just needs to be bled.
3) Low coolant level.
FIX: Check your coolant reservoir and/or radiator. Fill to appropriate level. Start engine, monitor coolant level in reservoir.
As long as your getting heat, then ANY of your fixes will be a DIY and not costly. Just act quickly because if it is a clog near your heater core and it is left over a long period of time, it can harm the heater core and potentially break it.