I understand the master key to be stored on the AxCrypt servers but it’s encrypted with your password. Obviously this means that your password is sent to the servers each time so you have to trust that AxCrypt aren’t retaining it and you also have to trust SSL integrity.
You can certainly backup the key pair using AxCrypt in case they go out of business or the servers go down.
Regarding your hypothetically compromised system… this troubles me too although it could be argued that if this happened then the attacker has access anyway no matter what encryption you use.
Obviously the old version of AxCrypt (1.7) works asymmetrically: enter a password and it’s done but regrettably it’s no longer supported or developed. I preferred the old model and, having scoured these forums myself, the best suggested alternative which is still developed, open source and free is BCArchive. It does all the same as the AxCrypt 1.7 but also has the option for symmetric encryption as well (like implemented in Axcrypt 2). It might be something you want to consider if you don’t like the new AxCrypt trust model (download here).