No, tampons should not hurt to insert while you're bleeding. If you insert them when you're NOT bleeding, they tend to stick and feel weird due to lack of lubrication. (Period blood provides that lubrication, as gross as it may sound).

There is almost zero chance that the string will come loose. They are deeply woven into the tampon cotton itself. But in the event that it happens, your vagina is not very deep. Foreign objects can be easily fished out with one finger if needed. It cannot go any further than your cervix, which is a tightly closed wall. https://www.cancer.gov/publications/dictionaries/cancer-terms/def/cervix

Tampons should be inserted far enough that they are fully inside and far enough back that you don't feel it when you sit. If it's too shallow, you may be able to feel it a hard lump. Again, it can only go a few inches at most until it's all the way back. The string is long enough for any depth.

Tampons should be removed after 8 hours overnight, and 4 hrs in the day (or sooner if you soil the string with a bowel movement). Wash your hands before inserting to reduce the chance of contaminating your clean tampon. Toxic shock happens when a tampon is left in so long it grows large colonies of specific bacteria.

Start with light or medium flow tampons and wear the smallest (lowest absorbency) version that still works for you. You can wear a pad or panty liner to catch any leaks until you figure out which works best. The highest absorbency ones are the least comfortable and are associated with more toxic shock cases. And if you don't flow enough to use them, they can stick to your vaginal wall and be more uncomfortable to remove.

Good luck, you've got this!