Basically having a climb speed means you can move at said climb speed after a successful climb check (and grants a bonus to climb checks) but doesn't let you do anything you couldn't do without a climb check.
Yes. if you can make the check to do so.
30 An overhang or ceiling with handholds but no footholds, or a typical city wall
Having a climb speed nets you a +8 bonus and you can cake 10, so with a +12 climb it's spiderman time.
A creature with a climb speed has a +8 racial bonus on all Climb checks. The creature must make a Climb check to climb any wall or slope with a DC higher than 0, but it can always choose to take 10, even if rushed or threatened while climbing. If a creature with a climb speed chooses an accelerated climb (see above), it moves at double its climb speed (or at its land speed, whichever is slower) and makes a single Climb check at a –5 penalty. Such a creature retains its Dexterity bonus to Armor Class (if any) while climbing, and opponents get no special bonus to their attacks against it. It cannot, however, use the run action while climbing.
seems kinda dumb if the creature has a climb speed equal to its land speed to have it move at the slower of the two in an accelerated climb.....If a creature with a climb speed chooses an accelerated climb (see above), it moves at double its climb speed (or at its land speed, whichever is slower) and makes a single Climb check at a –5 penalty.
It just means you cannot move faster than your normal speed while climbing, unless your climb speed is higher than your normal speed.