Can I trust this "Origin" DRM (Digital Rights Management) software platform?
Yes. It is fundamentally not worse than Steam as a DRM platform, although it doesn't have as many bells and whistles.
Has anybody done any independent verification on how this platform works?
Not that I'm aware of. "Independent" is practically impossible in a corporate environment, especially with a company as big as EA. Why would they let an outsider into their system, and how could any outsider they nominated be considered independent?
Does the "Origin" platform install anything particularly nasty or unwanted on my computer?
No, Origin doesn't install anything that a reasonable person would consider malware/nasty.