Can a failure in one MST instance can cause a failure in another MST instance?
The MST instances are independent. However, if a link fails, all the instances that are running on this link will be affected. So their fate is linked, even if the instances are independent.
Actually, in the IEEE specification, all the MST instances configured are always advertised on all the link (even an access port with only one vlan allowed on it would run up to 65 instances). In Cisco's implementation, only instance 0 (the CIST) is present on all the physical port. We only send the information associated to a given MSTI on a port if there is at least one vlan mapped to this instance active on the port. As a result, with our implementation, an access port configured in a vlan mapped to an MSTI X would only run the CIST and MST X (instead of all the MSTIs for the IEEE standard). If this port goes forwarding for example, only the CIST and MSTI X are generating a TC. So Cisco's implementation actually provides more "independent" instances that the standard and thus optimizes a little bit the consequence of a change in the physical topology of the network.
The answer is no.
The MSTP enables multiple VLANs to be mapped to the same spanning-tree instance, reducing the number of spanning-tree instances needed to support a large number of VLANs. The MSTP provides for multiple forwarding paths for data traffic and enables load balancing. It improves the fault tolerance of the network because a failure in one instance (forwarding path) does not affect other instances (forwarding paths).