Its no problem to move a GPU occasionally... but to do it every week is a whole other matter. The biggest issue is that PCI slots are not designed to be changed out that often. On a short term basis it would probably be fine, just understand that you are on borrowed time, and that something will break eventually.
On another note, it sounds more like the problem is that you need a more portable PC. There are some very capable small PC cases that can fit very nice parts inside without much thermal issue. Some even have handles on the top for easy transport. If possible I would consolidate your hardware down to one good machine in such a case, and just pack it with you when you need to move.
Thanks! I might just spend a bit of my money to get a new machine, i3-3220, 8gb ram etc and get a new case for that. My small case however, might or might not be able to hold the card.