Rama,
No, you cannot press on your cutting mat. There is one available which you can turn over to press, cut on the other. I think it is made by Harriet Hargrave. I think this style would be helpful if you are taking a class making you portable.
If you have been reading this forum, you will read about the storage of your mat. It should be stored flat away from heat and cold. Some of the ladies suggest hanging it with by a skirt hanger.
Just ask your questions here and we are eager to respond.
Blessings, Suzanne