I'm probably the odd one out on this, but I like the clutter. Gives a sense of overall feel for the job. Lot's of things to crosscheck and reference and paperwork to look over. Sometimes you can start to feel 'AHHHH!!!' with all the paperwork, but that's the job.
What I usually do is scan all paperwork for obvious mistakes (Date/Picture/Seals) I can catch. If I can't find any then I just handle one piece at a time. I'll look over the work permit since you just have to make sure the Seal is right, the name is right and the duration is correct. Once I know that's good I just hand it back to the immigrant and move on to the travel document and if that checks out, hand that back and then all I'm left is the passport.
Extraneous things like notes/cards/fingerprints given to me, once I have read or am done with them I put them on the far corner of the desk out of my way.
Like any task you just have to find your rhythm for doing things.
Also I've played 3 times now and have pretty much gotten the memory of city/country names. Since I think the game is going to be 30 days (is that correct) long, players will probably start learning them before things get too intense so you won't need to look in the book all the time. I still like it open on my desk. I usually keep the book in the background just in case I need it to highlight a rule.