Tampons are not haram in Islam. There are hadiths which indicate that it is permissible to use cotton to stop the flow of blood. For more, see the detailed answer.

It is permissible in Islam for women to go out during the 40 days of nifas to any place she wants, so long as she fulfils the Shar’i conditions (i.e., hijab etc.)

If a woman believes that her period has ended, then she starts bleeding again, if the blood has the characteristics of menstrual blood as described below, then this is menstrual bleeding, otherwise it is istihadah. If it is the former, she should not pray. If it is the latter, she should be careful and do wudu for each prayer, then pray.