Is there any way to have the HTC Logo on the HTC One (between the home and back buttons) to be used as a multitasking button AND press and hold for a menu button?
There are custom kernels supporting the HTC logo as a button. You might e.g. want to check tbaldens kernel @ XDA, which is available in different variants underlying the logo with different features:
- HOME2Wake and HTCLogo2Sleep
- HOME2Wake and HTCLogoMenu
Watch the videos over at XDA-Developers to see how it works. It is not the combination you asked for, but maybe that one was added later as a variant (another dev continued with a fork in this second thread, which lists a few more details), or you can ask the dev to do so.