For cleaning my airbrush I use Acetone, although Harder & Steenbeck did make claims of solvent resitance in this particular airbrush.
You 'should' break down your airbrush and clean it out after every painting session, then apply some lubrication as you rebuild.
I can't give any paasche specific advice, but my H&S brush does require a special tool for aligning the needle seal correctly. Its a screwdriver with an appropriate sized prong in the middle. To access the needle seal I have to remove the needle and the handle, and go in with this tool from the back. Paint builds up around it, so it does need a clean now and then. But DO NOT remove it if you can't realign it... your airbrush won't work quite right untill it's properly aligned!
Look into wether you need a special tool like me or how its done with paasche.
Sonic bath sounds good. I don't have one but always intended to get one, I'm going to write it on my shopping list now ... no more excuses!
For cleaning dried paint from the nozzle you can get nozzle reamers, ideally you want a paasche one with a taper that matches the inside of your nozzles.
For this I use an old well ruined paint brush with long bristles (a size 2 Rigger as it happens) with acetone, and only get the tip reamer out when theres real blockage problems.
If acetone isnt suitable for your airbrush; it should be fine on the chrome but potentially can ruin seals and orings.
Then try IPA (rubbing alcohol) or perhaps tamiya airbush cleaner.. or some other brand (I think tamiya is actually alcohol)- vinegar will corrode chrome which can be found in some glass cleaners check the ingredients
If you cant re align the needle seal, don't worry, just don't remove it untill you know how. As far as breaking down the brush goes you will want to remove the needle, nozzle, any caps or guards, and the trigger. I generally leave the air feed assembly alone.
When reassebling, apply a drop of airbrush lube (I actually use some skateboard bearing oil; bones speed cream) to the triggers moving parts, and use a clean rag with a drop on to oil the needle (wiping from the back to the point being careful not to bend it!)