Yes. They can climb up everything except ladders or a lip extending off of a wall.
Basically all of the solid and the liquid blocks, but they can't climb single slabs but they can climb double slabs.
Yes, spiders can climb almost all blocks. Use minecraftwiki.net if you have any more questions ;)
Spiders spawn at night and in dark areas.
no, some mobs can't, such as skeletons, creepers and zombies, but other mobs like spiders or endermen, climb over or just teleport in.
Yes.
You can't.
you get carrots in minecraft from baby zombies on chickens or spiders, you have to kill them and then you will get them.
you have to kill spiders
Currently, no.
Increases damage to spiders, cave spiders, silverfish and endermites.
It does extra damage to spiders and cave spiders, nothing else.
No, glass does not burn in minecraft. But in an odd occasion i put glass over lava and it started burning.