Continuing to do something just because it was done before, without knowing why

Solution 1:

These are two different things. You should not conflate them.

How would you describe someone that continues to perform and action solely because they have observed someone else performing that same action, but do not know the reason.

The colloquial expression that comes to mind here is "monkey see, monkey do."

Or, they perform an action in a certain way and do not change to a better method, only because they believe the current method must be the best way to do it because it has been done that way previously.

You could say they are set in their ways or hidebound.

Solution 2:

The term Cargo Cult could fit this situation. See the Wikipedia page http://en.wikipedia.org/wiki/Cargo_cult and scroll down to 'Other uses of the term'.

Solution 3:

At one extreme, sheeple can be used to describe people who follow the masses and act without thinking for themselves.