squeue to look at the list of running/queued or held jobs
sinfo to show which nodes are idle, busy or down
scontrol show node to get more detailed information on a node
For problem nodes - indeed just log into any node to see what a healthy node looks like
systemctl status slurmd
cat /var/log/slurm/slurmd.log
On your slurm controller look at the slurmctld and slurmdbd logs