Is a urine pregnancy test accurate a month after intercourse?

It can be if performed correctly. The test relies on hormones created by the pregnancy and they are often present at the point menses is due. The first morning urine is the more concentrated & thus more likely to be positive early. Such tests are more often falsely negative than falsely positive.

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.