Yes they do have claws on their paws.
All bears have the ability to climb trees. -- Many books will say that brown bears (grizzlys) can't climb trees. While they can climb if they have branches, the largest ones are too heavy to climb using just their claws, which are not as curved as a black bear claw is. All black bears can climb trees, but very large males may chose to run or fight if chased rather than flee up trees.
3 inches at most
Bears scratch trees to mark their territory.
Grizzly bears, brown/Kodiak bears and polar bears.
For hunting prey, and climbing trees. They also claw trees sometimes to mark their territory.
No
NO
No.
Because there are no trees in their area.
Most bears can climb trees, unless they are really big - like a full-grown adult male Grizzly...
OF COURSE bears climb trees, all bears are capable of climbing trees, especially the black bear. Others such as the grizzly or brown bear can climb up until the point that they are too heavy, then if they try to climb their claws just break off from the large weight. This is also a problem with the very largest of the black bears.