Will these "errors" in my advisor's letter raise red flags when I apply for academic faculty positions?
If I were your advisor, I would fix these typos before sending out any more letters (i.e., to other schools).  If there is an online system where I can instantly and unobtrusively replace a letter with a fresh copy, then I would do it.  Otherwise I would let it go.

These are not serious enough for you to worry about.  The impression that this will make on the reader is that your advisor is a human being, perhaps a human being who is rather busy.  Ideally your advisor will be someone whose reputation is known to the readers, so cannot be spoiled by a few careless typos.


  One of the positions uses the online application system where I am able to open the letters of recommendation and read them.


This concerns me...but you didn't ask about it.

Added: I missed at first this part of your question:


  [W]ill these errors give me enough reason to not to use my advisor's letter? I am contemplating not to list my advisor as one of my references, and I have read the responses to a question similar to mine in this forum, so I understand I am taking a risk and I will have to explain it if I choose not to list my advisor as one of my references. 


Heck no.  Not getting a letter from your advisor upon graduation is a very bad idea in almost all cases.