I have not done much traveling with my copaxone but I have done a little. I have flown with it. That part is easy. I just put the syringes in a plastic bag (I leave them in the blister packs) and set it outside my carry-on as it goes through x-ray. I have always included the precriptoin label in the baggy as well. Although copaxone says it can go a month out of the refrigerator, I only keep a weeks worth out. Everything else I keep next to a "blue ice" thingy. I get it into refrigeration once at my destination.
You may be able to ask for meds to be placed in a refrigerator on the plane for a very long flight but check ahead of time. You could also make your carry-on a small insulated lunch type deal with blue ice inside. Just open the whole deal when sending it through x-ray. If you have any questions, call the airline ahead of time.
Sometimes, my carry-on (usually it's just my large-ish messenger style purse) all I have is my wallet and my copaxone with the minimum of other junk. I feel a bit better carrying my meds with me, but I have only been gone for a few weeks.
I can understand your mom's concern. She's a mom. My neurologist said this to me (perhaps it will help with mom)........"there are no life threatening emergencies with MS. It just a matter of dealing with symptoms when they arise.) That made me feel much more secure about traveling. If I have a flare, I might well be miserable getting home but I am wiling to risk that.
Weigh the risks and then do what you feel you can. Try not to let MS limit you and the opportunities that may present themselves to you. China sounds very exciting!