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 ;)
no, some mobs can't, such as skeletons, creepers and zombies, but other mobs like spiders or endermen, climb over or just teleport in.
Spiders spawn at night and in dark areas.
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.
Currently, no.
you have to kill spiders
It does extra damage to spiders and cave spiders, nothing else.
Increases damage to spiders, cave spiders, silverfish and endermites.
No, glass does not burn in minecraft. But in an odd occasion i put glass over lava and it started burning.