It is impossible for the tampon to go anywhere but your vagina when you have sex. NO, it can NOT end up in the uterus unless something big is wrong with you. Just like your hand can not end up inside your friend's penis when you give him a hand job. If you don't believe me, take a mirror and flashlight and look at what is going on 'up there'. Or try to stick your hand into a soda bottle through it's neck. You will see that it is not possible.
The danger is that the tampon gets compacted at the top of the vagina, next to the cervix neck from the banging against it by the penis. This combined with the fluids can make the tampon pretty solid and can make it 'stick' in there. It can also dry out and make it even worse if it is left in. It is nothing that a bit of lube and fingers can't cure but many women are too afraid of their own vagina to do this. The doctor does not have anything special to remove a compacted tampon that you do not have at home. Just know that it can be difficult and a bit painful/uncomfortable and icky.
That said it is not a good idea to have sex with a tampon still in as it can get compacted and be uncomfortable to remove and if it bangs against the cervix, it can get pretty painful pretty quickly, just like a long penis with deep penetration can.
Yes, women are more prone to infection when menstruating because the cervix does not have its normal mucus plug and are more open. Combined with the fact that the uterus lining is in the process of 'dissolving' and 'disintegrating' this is a vulnerable time with germs present. As long as you and your partner is healthy and clean, no harm will be done. You anyway want to bath or shower before sex when on your period.
You can always have some shower sex. Menstrual cups are also an option. This way you also save the environment and does not have to use tampons that can pose a health risk on their own. Some brand names for menstrual cups are Diva Cup, Mooncup, the Keeper, Lunette, Ladycup, ..
Personally I enjoy period sex as much as my husband and I can't live without it despite all the naysayers.