An A319 Captain recounts a number of Controller errors which contributed to a loss of separation and a TCAS RA.

Narrative

ATC requested an expedited climb through 13;000 FT to clear traffic. Leaving approximately 11;500 FT ATC issued a right turn for traffic but used the wrong call sign immediately followed by a TCAS RA Descend command. Pilot not flying (First Officer) advised ATC we were descending to comply with the RA and they requested we not descend to avoid another target below us. Pilot not flying saw original target aircraft pass just above and to the right. TCAS reported 'Clear of Conflict' as both target passed behind our flight. ATC then cleared us back on course but issued the wrong fix requiring confirmation.A few minutes later ATC issued a frequency change but; once again; used the incorrect call sign requiring verification. No Safety Alert was issued by ATC during the event.To recount; ATC recognized the threat of the traffic but failed to ensure adequate separation. ATC further requested we ignore the TCAS RA. ATC must emphasize proper action; including the requirement for timeliness; when a traffic conflict is identified.