B737 FLT CREW CORED FOR TKOF ON RWY 6R AT CLE OBSERVED ACFT PASSING OVERHEAD WHILE TAXIING ONTO RWY; LNDG ACFT INITIATED GAR.

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TAXIING OUT ON TXWY L WITHIN 1000 FT OF TXWY P ON THE WAY TO RWY 6R; WE CHANGED TO TWR FREQ AND I HEARD TWR STATE AMONGST MANY OTHER RADIO CALLS (NOTE: FAIR AMOUNT OF TFC; OUR RELEASE STATED: EXTRA FUEL FOR HVY TFC IN CLE); 'YOU'RE 1/2 MI S OF COURSE.' NO ACFT RESPONSE THAT WE HEARD; LIKELY DUE TO RWY 6L HAVING A SEPARATE FREQ. TWR CLRED US FOR TKOF ON RWY 6R. AS PER OUR PROC; WE CLRED FINAL APCH VISUALLY AND OBSERVED NO ACFT AND TURNED ON ALL EXTERIOR LIGHTS EXCEPT THE RETRACTABLE LNDG LIGHTS AND THE STROBE. AS I ALIGNED THE ACFT WITH THE CTRLINE; TURNED ON THE OUTBOARD LNDG LIGHTS AND GAVE CTL TO FO; I SAW AN ACFT PASS DIRECTLY OVER US (HIS WING SPAN STRADDLED THE RWY CTRLINE) AND CALLED OUT TO THE FO; 'WHAT IS THAT ACFT DOING THERE?' WE HELD OUR POS WHILE I VERIFIED TKOF CLRNC WITH TWR. TWR CONFIRMED THE TKOF AND FOLLOWING MY QUERY ABOUT THE ACFT; STATED 'HE WAS CLRED TO LAND RWY 6L.' TWR GAVE DIRECTIONS TO THE OTHER PLT ABOUT CONTACTING THEM. WHEN I FIRST SAW THE ACFT IT APPEARED TO BE MOVING QUICKLY AWAY; A STRAIGHT WING ACFT. I THOUGHT OF A CITATION MOVING THAT QUICKLY; BUT THEN WHEN IT TURNED TOWARD THE N; IT TRACKED TO THE L RATHER SLOWLY. SUBSEQUENT TELECON WITH CLE SUPVR REVEALED IT WAS A MOONEY. GIVEN WHAT I SAW AND THE SIZE OF THE ACFT; MY PREVIOUS ESTIMATE OF 200 FT VERT MISS DISTANCE MAY BE OFF. COULD IT HAVE BEEN LESS? IT HAPPENED QUICKLY; MIGHT HAVE BEEN MORE; BUT I DON'T THINK SO. SUPVR SAID THAT THE TWR DIDN'T SEE THE ACFT; IT WAS NOT USING LNDG LIGHTS. BY THE TIME 'THE HORNS WENT OFF' THE ACFT HAD STARTED ITS OWN GAR. HE APOLOGIZED FOR THE INCIDENT; VERY CORDIAL CONVERSATION. QUESTIONS: 1) WHAT COULD WE HAVE DONE DIFFERENTLY? VISUAL PICK-UP -- WHEN I TOLD SUPVR THAT WE LOOKED ON FINAL BUT SAW NO ONE; HE SAID WE WOULDN'T HAVE SEEN HIM BECAUSE HE WAS BEHIND US. DON'T KNOW IF HE MEANT WHILE WE WERE HOLDING SHORT OR ON THE RWY. REGARDLESS; MAYBE I WAS LOOKING FOR LNDG LIGHTS VERSUS POS LIGHTS. HOLDING SHORT; SHOULD I HAVE LOOKED BACK AT 7 O'CLOCK POS VERSUS 9 O'CLOCK-ISH POS? 2) DON'T KNOW IF HE WAS A) STRAIGHT-IN; B) L BASE WITH AN OVERSHOOT OF RWY 6L; OR C) R BASE TO RWY 6L. BUT IF TWR TOLD HIM HE WAS X MI S OF COURSE (ACCORDING TO SUPVR PLT SAID HE THOUGHT HE WAS N OF COURSE; WAS DIALING THE LOC FREQ HEADS DOWN) THEY EITHER HAD A VISUAL AND THEN LOST IT; OR HAD A SKIN PAINT -- 'THE HORNS WENT OFF' -- WHY WERE WE CLRED FOR TKOF WITH AN ACFT SO CLOSE? I CAN ONLY ASSUME THEY WERE THINKING; 'HE'S CLRED TO RWY 6L;' IT WAS DARK AND NO VISUAL; WHO KNOWS. I CAN THINK OF ONE THING: AFTER MUCH THOUGHT; HE DIDN'T GO AROUND UNTIL HE SAW OUR EXTERIOR LIGHTS FLOOD THE RWY. WERE WE LATE ON THE LIGHTS?