Are la malplej juna and la plej maljuna entirely interchangeable? If so, which one is the most common?
Yes. They are interchangeable according David K. Jordan, in his book titled Being Colloquial in Esperanto. (I don't have the page number for this reference).
In the Tekstaro, plej mal- occurs 800 times. Whereas malplej occurs 67 times.
In Google, a search for "la plej maljuna" (with quotes) results in 50,400 hits. Whereas, "la malplej juna" (with quotes) results in 73 hits.
Based on this info, it is more common to use la plej maljuna instead of the other syntax.
No, they are not entirely interchangeable any more than a pair like the following are entirely interchangeable.
- He is the most pleasant one of the bunch.
- He is the least annoying one of the bunch.
This particular pair (malplej juna / plej maljuna) may lead you to point to the same person, but they are not completely interchangeable since they draw your attention to different aspects and so may have different connotations.
- La plej juna - the youngest one.
- La malplej maljuna - the least elderly one.
These have different connotations.