Docedison:
Why not drill the bottom and use self threading machine screws??? Mount it to an Oak or other real Hardwood and punk it up... Steam Punk Style also if you remove all that rubber stuff in the bottom you might.. drive a couple of screws up from the bottom to use as anchors for an epoxy bond if as I suspect there is a cavity under the "Rubber Bottom". Either solution I offered is IMO permanent... So maybe someone else has another opinion... Perhaps do it right and drill and tap mounting holes on the bottom of the unit... Lotsa solutions... If the bottom is 'perfectly' flat you could even use Shu Goo. I used it to stick a diamond lap to a piece of white oak wood for my kitchen knife sharpener after JB Weld failed twice. Lotsa Idea'sBob
Sooner or later, I'm going to have to upgrade my drill (handheld drill probably bought 20 years ago) or at least the drill bits. Right now, I don't think it would drill through metal as thick as the small base of my third hand unit. I've had some problems in drilling through oak. I also bought a concrete bit to mount some shelves on the wall in the garage, and gave up because I could not get more than 1/8".
In terms of the plastic underneath the third hand unit, the unit I currently have, does not have any plastic underneath, and there are no holes.
I was thinking of using my wood router to make a base to fit the third hand in, but I suspect if you knock it hard enough, it would come out of the base without having screws to hold it in.