NMAC. EVASIVE ACTION CLB. MULTIPLE RWY OP PARALLEL RWYS. SIMULTANEOUS ILS APCHS. WRONG RWY APCH.

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ON DSCNT INBOUND ON THE CVG 040 DEG RADIAL; CTR ASKED US TO CROSS 35 DME AT 12000 FT. WE WERE HANDED OFF TO CVG APCH; TOLD TO MAINTAIN 11000 FT AND TO EXPECT RWY 18R. WE DISCUSSED THE FACT THAT THE CLRNC TO 11000 FT DELETED THE PREVIOUS XING RESTR. FLYING WAS TRANSFERRED TO CAPT. I TUNED MY NAVAIDS AND SUBSEQUENTLY BRIEFED RWY 18R ILS. CAPT TRANSFERRED FLYING BACK TO MYSELF AND A CLRNC WAS RECEIVED TO 9000 FT AND SLOW TO 210 KTS. NEXT A TURN TO 270 DEGS; FOLLOWED BY A TURN TO 265 DEGS. NEXT WE RECEIVED A CLRNC TO 7000 FT AND SLOW TO 180 KTS. FREQ CHANGE TO 124.70; FOLLOWED; THEN A L TURN TO 210 DEG TO INTERCEPT THE LOC. WE WERE IN ICING CONDITIONS; I WAS DISCUSSING THE CONTINUAL WARNINGS FOR THE COWL ANTI-ICE AND THE MEANING OF THE YELLOW N2 VERSES YELLOW CAUTIONARY MESSAGES. I COULD HAVE MISSED THE RWY (ASSIGNMENT) AT THAT POINT BECAUSE I WAS DSNDING; SPOILERS WERE OUT; WE WERE BUSY. NEXT THE CTLR TOLD US WE WERE RIGHT OF COURSE. I PULLED UP TCASII ON THE MULTI FUNCTION DISPLAY. ALMOST SIMULTANEOUSLY WE GOT A TCASII RA CLB; CLB. I FOLLOWED THE TCASII AND WE WERE SEPARATED ONLY BY 500 FT VERTLY; 0 FT HORIZLY. CONTRIBUTING FACTORS I BELIEVE THAT WERE INVOLVED; TRAINING IN PROGRESS; INITIAL OPERATING EXPERIENCE FOR A CAPT IN TRAINING. ICING CONDITIONS AND CONTINUAL WARNINGS FROM COWL ANTI-ICE. COMPRESSED TIME FRAME AFTER HDOF FROM CTR UNTIL INCIDENT. I BELIEVE THAT WAS REALLY CAUSED THIS INCIDENT WAS THE FACT THAT WE WERE BOTH SO BUSY THAT WE WERE NOT PAYING ENOUGH ATTENTION TO WHAT THE CTLR WAS SAYING; COUPLED WITH THE FACT THAT I WAS CONDUCTING SOME TRAINING AT AN INAPPROPRIATE TIME IN THE DSCNT. TO PREVENT A RECURRENCE; PERSONALLY I WILL REEVALUATE MY LISTENING ABILITIES AND RADIO PHRASEOLOGY AND THE DIVISION OF DUTIES BELOW 10000 FT.