Q: Can I leave my switch ports in auto mode?
A: The questions do not specifically ask you to configure speed and duplex but you should do this to avoid any mismatches that could cause you problems. It might cause your switch to enter in a loop and stop your entire network.
Q: Should I name the VLANs as VLAN_11, and so on?
A: If the question named the VLANs, you should configure them as you see in the question.
Q: Should I be able to ping the switch's loopback?
A: Unless you find a specific requirement on any question, you are not required to accomplish this.
Q: I think after I have configured the vtp connection, I should be able to see all VLANs in both switches, but I do not. Why?
A: Yes, you are right. Please double-check your configuration. You could be missing something in your configuration.