I was wondering if whippet could be used as a trekking pole? Question arises from hiking on steep hills (no snow, dry grass), where any additional grip would be really helpful. I've read that most people offer to use ice axes in this kind of situation, but I was thinking, if I already use trekking poles, wouldn't I be able to accomplish two things (stability while walking and self-arrest/assist) if I'd substitute trekking poles for a whippet?
I don't see why you couldn't. One key difference between ski poles and trekking poles is trekking poles often have a little shock absorber in them and ski poles never do. If you don't care about the shock absorber then you can use a whippet as is. If you want the shock absorbing properties of a trekking pole and the pick of a whippet then you might need to modify your trekking poles/whippet pole.
The easiest solution is to unscrew the handles and swap them out if that is even possible. I have a very early model whippet, and the pick is actually a metal insert that can be removed and replaced with a spacer when I don't need/want to risk falling on the pick. It would be fairly easy to modify a trekking pole to accept the metal whippet insert. The whippet in the picture you provided looks much more polished than mine so it might not be removable without destroying the pole. If you're willing to destroy the pole then I'm sure you can affix the whippet to a trekking pole even if it is no longer designed to be removed.
On TetonAT.com I found a 9 year old post that shows modifications being done to the old style whippet I have. I don't know how much of it is applicable to your needs with the current generation whippet so I won't pull any of the pictures in. In the comments though they mentioned that a future whippet will be affixed to the grip with rivets. If that is still true then you could drill out the rivets and use screws to attach it to your trekking pole like Steve did in his article.
... the ultimate ski mountaineering tool ... that can help keep minor slips from turning into slides for life.
For ski mountaineering, it is not feasible to carry an ice axe in your hands so the Whippet is better than nothing. Performing a self arrest with an ice axe is difficult and I would have serious doubts about the ability to effectively use the Whippet for self arrest. The four issues I see are (1) the blade appears to be shorter than on a typical ice axe, (2) the strength of the shaft of most trekking/ski poles is substantially less than that of an ice axe, (3) the position and length of the leash may interfere with maintaining control of the "axe" and getting proper hand position for arresting, (4) and the quality of the spike.
I think overall a mountaineering ice axe is better suit for self arresting than a trekking pole with a pick. That said, self arresting on grass may be sufficiently different.