As-salaamu `alaykum.
May Allaah assist you in your affair, and keep you firm and strong on the deen. What is happening is a trial/test from Allaah, so patience is best. With regards to the prayer mat, then it isn't exactly "Islamic" anyway. It's merely a mat used by Muslims to pray upon. The prophet sal-Allaahu `alayhi wasallam would pray on bare sand, mud (even when wet), palm leaves, and other stuff. So inshaa' Allaah, if you feel your floor is unclean to pray on (in your room), then just lay a shirt or something.
As for moving out, then eventually that may be the only option, but we hope that they soon take shahaadah, or at least accept you. It is of course best (in you have to move) to stay with a relative who is far more tolerant and maybe open to Islaam, because in the near future you're outward Islaam may affect them positively (which alhamdulillaah, happens much), whereby they will "approve" of you more. This in turn can help your scenario with your parents, but that's of course in the long run.
Hope it helps,
was-salaam