Looking at the zombie template it states that a zombie has no skills. Can a zombie still make skill checks like jump, climb etc?
Yes, a zombie can make (untrained) skill checks
Having a skill simply means that the monster has ranks in certain skills. The monster block, then, only lists those skills (or other interesting skills, like one it receives racial bonuses from). After all, listing all skills for every monster would take a lot of space (and be unreadable).
As a mindless creature, a zombie has no skill ranks. It also has no racial skill bonuses. Therefore, no skills are listed.
However, creatures can always try skill checks if they're physically or mentally able to do a thing, as long as the skill allows untrained checks. They'll use their ability modifier as normal, without a skill bonus. A human commoner zombie, then, would have a climb and jump modifier of +1.
I'd agree that they can make any normal physical skill check (climbing ect).
Draw the line at whatever makes sense. Zombies have no con, so running doesn't make sense (if they can run, they always run at max speed, no endurance/con checks).
Other things like perform are untrained, but zombies don't exactly act well.
Go with what makes sense