ZID Controller described a separation event during a busy traffic period which included aircraft with similar flight numbers; a military flight requesting a break up and an aircraft reporting smoke in the cockpit; all leading to a missed clearance instruction.

Narrative

Air Carrier X northeast bound FL370 turned 5 degrees left to go behind Air Carrier W and Air Carrier Y. Air Carrier W turned 5 degrees right to stay in front of Air Carrier Y. Air Carrier Y assigned present heading to go between the two. Air Carrier Z blocked the transmission to Air Carrier Y asking if the turn for Air Carrier W was for him. After correcting the confusion I had another flight at FL350 declare he needed a different altitude. There had been chop in the area so I assumed he wanted to change altitude for the ride; he wasn't very clear as to the need for change nor was the transmission legible. He stated again he wanted lower and this time said smoke in the cockpit. My attention was diverted from the Air Carrier Y missed readback as I assisted the aircraft with the smoke issue with alternate airports and getting him down to 10;000 FT. Conflict alert activated and TCAS was invoked by Air Carrier X. Air Carrier Y said he never was assigned present heading when questioned. Also there was a military refueling flight requesting to break up at the same time. Recommendation; better radios and I was told; that the similar call signs Air Carrier Z and Air Carrier W had been turned in numerous times.