Is it appropriate (as a PhD student) to email other researchers asking about some details in their papers?
As a professor, I've received many emails from graduate students at other institutions over the years asking me questions about my research. Not once has any of the following thoughts ever crossed my mind following such an email:
This student is doing something "inappropriate" by contacting me at my publicly listed email address.
This student is "stupid" or is asking a "stupid question".
Oh, this student mentioned their famous advisor. Their ignorance reflects poorly on their advisor. I will make sure to complain to the advisor the next time I see them.
Oh, this student didn't mention their advisor. Something's fishy here. I need to know who their advisor is, and preferably know it's someone famous who can help me in my career, before I dignify their email with a reply.
Etc etc, i.e. any other bad thought that suggests the student has crossed some invisible line separating lowlifes like them from (supposedly) important, powerful people like me.
Instead, here is a sampling of ways I actually react when I receive such an email:
Oh, how nice. Someone at [name of cool university] is thinking about [name of my paper]!
Oh, how nice. A student of [name of famous advisor I am friends with] is thinking about [name of my paper]!
Oh, how nice. This student pointed out a weak point in the presentation of one of my proofs. I hadn't thought of the question that way, it's quite interesting! Seems like a smart student, would be nice to meet them some day.
Hmm, I'm kind of busy with lots of other things so this email stresses me out a bit. Oh well, I'll find time to reply to it some time in the next few days. In the meantime, how nice that someone is taking an interest in my paper and has interesting things to ask/say about it.
As you can see from these two lists, basically the whole premise assumed in your question about the mindset of a professor receiving an email of the type you are thinking of sending is deeply misguided. Professors (the vast majority of them at least) are simply nothing like the scary, classist, pretentious, easily offended, thin-skinned people that a lot of students and other people seem to think they are. They are busy, yes, but the main thing that makes them so busy is that they are passionate about their work and usually cannot pass on an opportunity to discuss it with someone, regardless of their academic rank. So, good luck sending your emails! I'm sure you will be pleasantly surprised by the replies you get. :-)
I think it's perfectly fine. They publish their research because they want other researchers to know about it. Maybe they will ignore your question, or maybe they will answer - in any case, it's not weird.
If you are worried about making your advisor look like an idiot, I would ask him/her the question first. If they cannot answer it or they do not have the time, then, by all means, reach out to the corresponding author of the paper in question.
More importantly, you shouldn't let your advisor's popularity get in the way of you doing great research, so don't do that.