How should I maintain my trekking backpack? Are there things I should do/regularly? How should I clean it, if necessary? How should I store it?
This question is related: Taking care of hiking backpack
Personally I have not found backpacks to be very high-maintenance. After a trip I completely empty my pack, shake it out, and wipe off the dust with a damp cloth. If there's sap or other problems I'd try spot cleaning them with mild detergent, but so far I've been lucky. One thing I'm careful to do (with tents and other gear as well as packs) is to prop them open and allow them to thoroughly dry before putting them away. (The last thing you want is for mildew to start.) After much use, or in excessively sandy/dirty areas, you may wish to treat the zippers by brushing off any debris and applying some form of zipper lube (spray or wax); this will help extend their life.
Storage, as with most things, should be a cool dry place away from sunlight.
Osprey suggests washing their packs every so often; you can use a bathtub with warm water and a mild detergent to do this. Gently agitate the pack, rinse thoroughly, and again make sure it dries!