In my smallest Veiled vivarium I use bamboo sticks and pothos vines for climbing.
In my ellioti viv I use one long fake vine from I believe exoterra, pothos vines, sticks I found outside and a nice bushy umbrella plant.
With my largest veiled I use a fully grown ficus tree that has thickish branches, an exo terra vine, pothos vines, sticks from outside and an umbrella plant that still has to grow.
As you can see, my favourite is the pothos. They grow quickly, are easy to maintain and the chams love them! Every cham (except my largest veiled) sleeps on a select pothos vine. They're the perfect diameter that allows the chams to properly grip without slipping. I would recommend at least one pothos to every cham owner! The best investment I made was last year when I bought a small pothos for 2€ and gave it to my husband's grandmother. The plant was, after almost a year, more than long enough to fully vine out a 60x60x120cm vivarium! They grow REALLY quickly.
Properly sanatized sticks from outside would be my second choice for natural sticks. Bamboo is also great, but I find it looks too unnatural. If you think it looks good though, they're also really good. Just be sure to get the right thickness for your cham as the bamboo can be slippery
::EDIT::
For the mist drops I have the same problem. I am trying to figure out how to mist the new viv best without getting a lot on the floor. For now I have some towels that I leave on the floor and one behind the viv. They get changed every couple of days as they never get TOO wet. If anyone has any ideas... I'd also love to hear them!