1 and 2-yes. all nepenthes climb except campanulata and rarely parvelii (sp)
number 3-let it hang down if it's in a basket (it will flop over, then start growing up, over and over), let it sprawl, stake it, or cut it back and make cuttings.
they also make basal rosettes when they are big enough.
also, their tendrils on the upper pitchers grab whatever they can grasp , just like a grapevine, and this is how they support themselves in the wild. they twine over and around a stick/limb/ plant/whatever. if the tendril doesn't touch anything, it winds and coils aroudn itsself
there is also a form of veitchii that hugs a treetrunk with it's leaves. (litterally, it wraps them around it and it looks like it's hugging it)