How long does sperm survives in the vagina? is the first day of sexual intercourse is counted as day one?

Up to a week: for highest chances of conception, 2-3d from ovulation though. and sperm would not hang out in the vagina - it would have travelled to the fallopian tubes.

Sperm survival: Sperm can survive inside a woman's body up to 6-7 days under the right conditions.first day of intercourse would be day 1

Several days: Sperm can severe on the order of three to five days. Timing does start when the sperm is deposited in the vagina.