I usually use an insulated pair of pliers to remove my anode caps (the correct term for the "suction cup") from CRTs. I carefully use the pliers to pinch one of the two pincers inwards and then carefully pull up to remove it from the hole on the CRT.
As far as discharging--from my personal experience, you should have nothing to worry about if you are working on a Classic (I or II, either analog board), even if you just switched it off and pulled the power cord. They discharge as soon as the machine is switched off. I've never felt anything through my pliers or fingers when operating on a Classic. The same can be said about later SEs, although I've heard the earlier SEs didn't self-discharge. I've done CRT transplants on SEs from 1988-1991 and have never had a problem with discharge. My guess is that Apple added the self-discharge function when the analog board was redesigned to accommodate the new fans.
If the SE has a "rat cage" fan, it's probably a wise idea to discharge the CRT. This can also be said about any Plus or earlier model. Of course, if you wait a long time, the CRT will lose its charge as long as the Mac is unplugged. (I replaced a CRT on a Plus after it had sat idly for ten months; this machine had lost its charge in the CRT by then).
Granted, some folks do prefer to discharge every CRT regardless of computer model or length of time since the Mac was last plugged in. Most manuals will tell you to do this and so will those who always err on the side of caution. However, these discharge steps really aren't necessary with the newer compacts--I've worked on dozens of them without using the discharge procedure and have yet to be shocked.
(I should point out that I have yet to perform a Color Classic CRT replacement. If anyone has experience with these let me know since I want to replace a CRT in a Color Classic sometime soon and would like to know if I need to dust off the discharger).