I've done a ton of monitor work the past few weeks, and I can tell you that some monitors make a very satisfying 'SNAP' when you discharge them, and others make a small 'tick' like if you rubbed your feet on the carpet and touched someone. There's really no way to tell whether or not the tube is still holding a charge, so always better to be safe (in your own mind) than sorry.
For discharging, I used to use a very long (but narrow tipped, to fit in the anode hole) screwdriver with jumper cables from your car attached to the metal shaft, and attached to the metal frame of the monitor. I have since gotten a better made setup (from Bob Roberts, thanks!) that works great.
You need not worry about touching the anode cup if the monitor is off. All the juice (hence the PAIN) is located in the tube itself -- it acts as a big capacitor, storing charge. Once you've FULLY discharged the tube (touch a few times repeatedly, over the course of 5 min to be sure), you should be good to work around it for a little while. But if you've let the monitor sit for even 30 min to a few hours or days, it's a good idea to discharge it again. Some tubes have a tendency to build a small charge again over time. Not as much as the initial charge, but enough to get your attention.
I'm always super cautious. Overly so, in some cases. But yes, you can fold back the suction cup and work the anode clip into place once you've successfully discharged the tube. Make sure both 'teeth' of the clip are clipped in place. I always wiggle/spin the anode cup a bit to make sure it's firmly in place before powering up the monitor.
If you want to be even more cautious, keep your left hand in your back pocket while you discharge AND while you hook the anode cup back up again. That will lessen the chance that any electricity will find a path to ground through your body.
Good luck!