It didn’t in the stock drivers at the time I wrote that, and I believe it still doesn’t, but some community members have written one or two drivers for some specific Zigbee multiendpoint switches which are mostly working (with some glitches), and The Smartest House is said to be working on some for some of their zwave models. But as always, the model number matters. Here’s one example:
- Is there a centralised list of device drivers for Edge?
Try this:
Edge Drivers Section Added to Wiki Quick Browse Lists
And There are discussion threads here:
Writing Edge Drivers - SmartThings Community
- Is there any possibility of crowd-funding (or funding in any way possible) the writing of a device driver?
Sure, many community members have patreon or other contribution accounts. You can start a thread titled: “Edge Developer needed: Name of Device” in the community-created DTH section and tag it edge-device and describe what you’d like and see if you get any responses.
- Is Groovy going to be killed off completely? In other words, is there a cut-off date when Groovy device handlers will simply cease working, rendering my 40+ switches completely dumb?
That’s actually 3 different questions, and they haven’t publicly announced specific dates yet. 
What’s going away is the Samsung-provided free groovy cloud. If you wanted to host your own groovy server and wrap an Mqtt or other interface around it you could, but it would probably be just as easy (and much more efficient) to write a new Edge Driver instead.
But yes, at some point the existing custom self-published groovy DTHs will stop working.
- I have a ton of piston code on webCoRE - what will happen with that once devices used in webCoRE are ported over to Edge (whenever such device migration is possible)? Will the pistons stop working?
Webcore is a groovy smart app, hosted in that same free Samsung cloud. So it dies when the cloud goes away, regardless of what DTHs you use. Start preparing. 
- Is there any real alternative for webCoRE that runs locally?
Too big a question: there will be multiple official alternatives (the rules engine built into the SmartThings app and the official Rules API) as well as the ability to write and host your own smartapps. Some people are adding a Hubitat hub just to be able to use Webcore there. I suggest you check the webcore forum, I’m sure there are discussions there.
Or you can start a new thread in the webcore section of this forum, maybe something like “The Future of Webcore after Edge?”
These are all very good questions, but way off topic for this thread, so please continue discussion at the various links I’ve provided.