i believe you will be doing some pub crawling in the düsseldorf altstadt (old town), this is definety a casual place. although düsseldorf in general is posh, the altstadt is not posh at all!

Does that mean we'll get into clubs wearing trainers?

I don't want to be turned away :S

Will we require formal shoes or can we wear trainers in the Alstadt and nightclubs? Are they strict on dress code?

Less strict and the Alstadt is the place to go. I believe wearing jeans will be no problem with trainers.

some hip locations requite smart casual style

(for example Jase in the "Altstadt" or 3001 in the media harbour)

I would suppose this depends very much on what do you like...

If you go for nicely dressed girls you are suppose to dress accordingly. If you just want to hang out with the bunch in the Altstadt and do a pub crawl and get very drunk - anything goes ;-)

At least you have the chance to meet well dressed people which are still open for a small talk - you will visit the Rhineland - people here are great small talker...