Will pregnancy test work at around week 13?

A properly done urine pregnancy test will be positive 2-3 weeks after exposure and thereafter. A test can be falsely negative early on and if done on dilute urine. The first morning urine is the more concentrated & likely more accurate sample. By 13 weeks the hormones that turn tests positive are in constant production.

Yes. Conception occurs within day or so of the sex. It takes about 3 weeks for the embryo to make enough HCG for a positive pregnancy test. Blood test may be done a day after a missed period. Home pregnancy tests should be done 4-5 days after the missed period. If the result is negative repeat the test in 4-5 days. Use first morning urine and follow instructions for the test carefully.