1. Monitor DST10, from Help topic DMD0208:
contains a value that reflects the current operational mode of the controller, 2 = STOP mode, 3 = RUN mode
2. The first 16 discrete inputs in your ERM configuration show the Slave Status Bits and whether a slave is in error (ON means OFFLINE or IN ERROR). Say you map it to X300 in ERM Workbench, that will be DLX300 in Do-more. If you want to see if ANY bit is on, cast it as a Word and see if it's non-zero, e.g. DLX300:UW (or look at each bit individually DLX300 for Slave 1 in error, DLX301 for Slave 2 in error, ... DLX317 for Slave 16 in error).
3. No. If you want your HMI to control whether your PLC is actually running, make that be a C bit that is a huge "disable" throughout your ladder logic program, then turn the C Bit ON or OFF from your HMI, with the PLC always in RUN mode. Run mode is HUGE safety consideration (some PLCs use mechanical keys to "lockout" mode changes via programming software, not even HMIs).