Assalamu Alaikum
Mashallah sis, I think it's wonderful that you are so into your deen and are perseverant. May allah only increase it for you and make us as eager to please Him as you are ameen.
Is your grandma a Muslimah? Until you find your Quran or get it replaced, you can read it online:
https://quran.com/
http://www.quranflash.com/books/Medina1/?en
http://www.clearquran.com/
Also, if you have a smart phone or tablet, you can download apps.
As for your prayer mat, you don't necessarily need to use one for prayer. You can just use a clean towel or cloth. I'm sorry that you are facing difficulties with your family, but be patient and stay strong. Inshallah allah will ease things for you and guide your family as well.