Q:

Two men and a door, but no liar

You come across a door, guarded by two mysterious men. As you start to figure out how to tell which one is lying and which one is telling the truth, they say in unison:

WELCOME, TRAVELER, TO REIBELLO'S RIDDLE REPOSITORY.  SHOULD YOU SEEK ENTRY, YOU'LL HAVE TO SOLVE A SIMPLE PUZZLE.

You quickly agree to hear the puzzle.  You also notice that their voices are different.  One is deep and strong, while the other is softer, and strangely emphatic about his words.

Our first AND second facilitate, that which you need to operate.
Each alone, is not correct, but read aloud make our subject.
Our third is imprecise at best, but gladly taken on a test.
Our fourth is five- It's two.  His clues are the worst.  It's also half of what is first.
Our fifth is fifth.  All the clue you'll need, I reckon. But just in case, it's also our second.
Our sixth and last, core of our planet round, but don't go quite as far in underground.
Our riddle complete, answer if you dare.  Six letters, and no more, WE SWEAR.

What word are the mysterious men looking for?

A:

The mysterious men are looking for the word:

 WEAVER.

Our first AND second facilitate, that which you need to operate.

 WE (not sure)

Each alone, is not correct, but read aloud make our subject.

 ???

Our third is imprecise at best, but gladly taken on a test.

 A. Imprecise ???. Anyone would be glad to receive an A as a test score.

Our fourth is five- It's two. His clues are the worst. It's also half of what is first.

 V. V is 5 in roman numerals. V is also half of W.

Our fifth is fifth. All the clue you'll need, I reckon. But just in case, it's also our second.

 E. E is the 5th letter of the alphabet. This is the same as the 2nd letter.

Our sixth and last, core of our planet round, but don't go quite as far in underground.

 R. Middle letter of Earth. Also appears in underground

Our riddle complete, answer if you dare. Six letters, and no more, WE SWEAR.

 WE SWEAR = WE AVER

