Here is a lowdown for you on STI testing, and what tests for each involve:
Testing, Testing.... You'll note that none of those tests are urine tests, with the exception of Chlamydia, which can be tested via urine. Urine testing is mostly for pregnancy tests or to look for urinary tract infections only when it comes to reproductive/sexual health. I'm not sure why your doctor did a urine test or what they were testing for, nor why, if they did, they gave you the idea that was a way to test for STIs. Sounds like there may have been a communication breakdown here, so for future reference, what you just want to do anytime a doctor says they are doing something to test you for something, you want to ask what they are testing for, specifically.
I can't know your mother's STI status. If she has never had any sexual healthcare at all (which is unlikely, especially if she got any care while pregnant with you or during delivery), she can't, either. HPV, however, is not one of the STIs at all likely to be passed to an infant during birth. HIV and Herpes are the ones to be concerned about there. Mind, the vast majority of people who have been at all sexually active in their life will have contracted or will contract HPV at some point. Not everyone, but most. When people use barriers for intercourse or oral sex, it is way less likely, but even then, barriers do a better job of protecting against infections that pass via body fluids than those, like HPV and Herpes, that pass via skin-to-skin contact, because condoms (unlike dental dams) do not cover the whole surface of either person's genitals. Too, don't forget, if you have not had them already, that there are now immunizations for HPV that do a great job of preventing the most strains that are actually dangerous: those that can result in cancers. If you haven't had those, that's something else you can ask your doctor about.
The terms STD and STI are often used interchangeably, however, for most of these infections, STI is more accurate. The I = infection. The D = disease. And disease, on the whole, is a term used medically to describe illness that is necessarily progressive. However, as most STIs, when treated, are not, most are not actually diseases, but only infections, thus, STI. Make sense?