Evo said:
Bioquest, a tampon is inserted into your vagina, which is were the menstrual bood will come from. Your urethra is near the vagina, so when you urinate, the menstrual blood will mix with the urine and appear to have blood in it.
Correct insertion of a tampon into the vagina will not interrupt urination.
Are you close to your mother or other female relative or teacher that you can talk to about menstruation?
Here is a link that will show you the female anatomy. You can see how close the urethra (for urine) and the vagina are.
If you don't have anyone close to you that you feel comfortable discussing such things with. Tell your doctor that you need basic information on your urinary tract and menstruation.
Let me find a less graphic example.
Even the illustration Evo provided makes it look like the openings are much further apart than they are in reality. Imagine a coffee stirrer taped inside the top of a garden hose...the coffee stirrer would be the urethra and the garden hose the vagina and the tape the amount of tissue separating to the two "tubes."
Menstrual bleeding will end up in a urine sample if you do not have a tampon inserted, just because the urine will "wash" blood off the exterior of your vaginal opening. If a woman has her period and it is important to check for blood in urine (i.e., if she is in the ER after a serious accident), the sample is usually collected using a catheter to avoid this problem. From home, the easiest way to do it is use of a tampon.
If you've never used tampons before, you can put some vaseline or similar product on the applicator if you have trouble inserting it. Insert it upward and slightly angled toward your back (it can help to stand with one foot on the toilet or edge of a bathtub), hold the wider part of the applicator, and push in the narrow part to insert it deep into the vagina. Remove the applicator, wash yourself up, and take your urine sample. If you do not want to continue using tampons, you can remove it then. Perhaps you have a female relative or girlfriend who uses tampons who can give you one so you don't need to buy a whole box. For a first-time user, I'd even recommend getting the slender or light ones, just so it's easy to insert, especially if you don't plan to keep it in long or continue using them.
Edit: Keep in mind, my suggestion here is ONLY to help you convince your doctor there is a real problem. If your doctor is still dismissing you, and you are still experiencing symptoms, stop messing around with making appointments at the wrong time of the month to be useful to convince them, and take yourself to the ER on the very first day of your period when your symptoms are the worst so they CANNOT ignore the problem. As we've repeatedly told you, there is nothing normal about what you are describing, and you MUST get medical attention.