How to deal with errors in well established papers I am currently supervising a Bachelors student for his thesis. I gave him a well-established paper to read of which I only knew the results but never checked the calculations in it. It was first published in 2006, is currently in its 6th version on arXiv and has about 330 citations, so I expected it to be mostly without mistakes...

Well it turns out I was wrong. My student stumbled over several mistakes and I found some more while trying to explain some calculations to him. In total we have found about 10 errors - some minor like a wrong index in a formula, some major like claiming that a function is convex while it is in fact concave and an inverted inequality sign in the final result...

Under different circumstances I would write the author, but I feel somewhat silly writing an email that basically says "Hey, this paper you wrote 9 years ago contains some errors." I really cannot imagine that we are the first to find these mistakes...

I also thought about commenting on the paper on https://scirate.com/ - this way the author needs not bother with it, but following generations could still see the corrections. On the other hand this could be a big insult to the author...

Is there any other way to make sure following generations don't stumble over the same mistakes? (The results are basically so well established, that active researchers take them for granted and don't try to verify the calculations - but following students will surely stumble over the wrong equations)
Should I just drop it? Should I still write to the author?
What is the "proper (scientific) way" to handle this situation?
The most important thing, I think, is that the core results stand.  This means that correcting the errors is not about substance (which might require a new corrective paper or a retraction, depending on the nature of the errors), but is instead about making the road to understanding easier for others who also read the paper.

Since you say the paper has been revised multiple times on arXiv, it seems there is a decent chance that the author may still be tracking on it and be willing to update and correct it.  I would thus start by writing to the author---it's entirely possible you're the first to notice the mistakes, simply because everybody else thought they saw the things that were supposed to be there (copyediting equations is hard).  If that doesn't produce a result within a few months, write to the journal and see if you can get the errata corrected in their version of the paper.

Beyond that, I think there is not currently a high incentive to put in further effort or to put material up on a third-party website, simply because I suspect that most people who look for the paper are unlikely to readily encounter such annotations.  Search engines are pretty useful, though (as this site readily attests), so there's no reason not to put the corrections online somewhere as an FYI to help others who might happen to search for "errors in Paper X."