Well, if there is not BL on the Mini, and I assume there is not and that's why you are trying to burn a new one, then you will not be able to load a sketch onto it using the BL (through the on board UART).
So, getting your USBasp working would be the first step. Have you tried it though a different USB port, or on a different PC?
If there is a BL on it, the the error you are getting trying load a sketch might because you are not resetting the Mini at the right time.