Yes. But not with commonly available equipment.
No idea. Perhaps you need to talk to a marine biologist or a bat scientist or something. Someone who knows about sonar would know all about this.
How would I know if it worked?
By using some kind of sonar receiver and an oscilloscope perhaps.
would meters in Sound Forge move?
It would depend on some specifics, but I would doubt it. I'm thinking in terms of limitations of the file format (does it even allow encoding of HF data?) and the DSP.
Would playback through speakers be able to reproduce it?
Would depend again on the speakers in question, but in general I'd have to say no. It would also depend on exactly how high you're going. Like, 30khz might be present, but 100khz will be so attenuated that it wouldn't register.
you would need to "feel/touch" the speaker itself to determine if movement was present and hence a sound was being produced
Even so, there'd be so little energy present that you might not even be able to feel it.
This is certainly an interesting question, Introvox. Why do you want to do this? More info might help us come to some sort of an answer for you. What is the sound source you wish to record?! That's got me intrigued!
I'd start by looking into ultra-sonic technologies and sonar systems. You can buy ultra-sonic transducer fairly easily (car reversing sensors etc.) and you could easily buy or build a tone generator that would create a constant ultra-sonic tone. But whether that suits your needs or not...
Certainly, a standard audio workstation won't do. The sound cards are filtered and the DSP has anti-aliasing filters that would wreak further havoc, and the file formats might not even allow data to be encoded to represent such high frequencies.
Common systems would treat it as interference and block it out or ignore it. You need a specifically designed system to record it.