Federation Commander

If one is using the optional rule to take a frame damage hit instead of a "skip" on the DAC, does the ship go "boom" as soon as all the frame boxes are hit, even if there are internals still undamaged? (There will be, definitely.)

The ship goes boom on the frame damage hit after the last frame damage box is disabled. So, if a ship has four frame damage boxes it takes the fifth to actually blow the ship up.

To actually answer the question, yes. If you accumulate enough frame damage using (3E1a), you will explode regardless of the number of internals remaining.

Do note that using (3E1d) is an exception. If using that rule to protect a "last box", you cannot cause the ship to explode directly. But, of course you will still likely explode sooner doing that.

Is the rule about taking a Frame instead the last operational box of a system an optional rule or just an option for the player taking the damage?

It is a rule that provides players with an option. So, the second.