Can hCG be produced the day after intercourse? Would a urine home pregnancy test detect pregnancy this soon? If not, how soon?

No. Hcg can be detected by blood test at day 11 after conception and day 12-14 by urine test.
Pregnancy tests. Depends on the test. Many within the first two weeks. Most doctors recommend you wait until the first day of your missed period since the HCG has to build up. Urine tests are 97% accurate. Blood tests pick it up sooner and are more accurate.

Related Questions

Would a blood test be any different than a home pregnancy test at this point? (47 days after intercourse 2 months no period) HPT showing negative

Blood test better. Blood tests are considered more accurate, That's not to say the home tests aren't accurate. More to the point, a blood test can give both a qualitative answer, (yes or no), but also a quantitative answer, (is the pregnancy advancing as hoped).