1) The Guards can have a benefit when implemented wisely. But on the infrastructure you should only set it when you know how your infrastructure behaves. On User-facing ports, using root- or BPDU-guard is probably a good idea.
2) It's ok that way. The preferred root has the best priority, your next preferred device if the first one fails gets the second best priority. All other switches can typically stay at the default or any value that is larger than the first two.
3) It already should be if there is no other STP-device in your network. The switches will tell you which one is the root. How did you test it that you say the core is not the root?
3) I clicked on the core switch and its listed in the Summary tab. The RSTP ROOT has the other switch listed with a higher bridge priority. I then clicked on that switch and noticed RSTP ROOT has the value "This switch".