In Seminary, we translated from the Masoretic texts. I know in Ph.D Greek courses, they translate the LXX. So yes, some differences appear. I have also spent some time studying the LXX in Greek, as well as comparing words to the ones that appear in Hebrew, and the Greek New Testament. But none of the differences affect doctrine. I know Jeremiah, the LXX takes away a lot of things. Hence, most translations use the Masoretic, although the things that are missing are mostly "The Lord says" and things which are not doctrinal in nature.
As far as the Greek texts, the same thing applies. The differences in the manuscripts are so tiny as to not affect doctrine. If you learned Greek, you could get a UBS or Nestle-Alland NT and all the textual differences are written below the text for every difference. After a while, you begin to see how these differences do not affect the text.
As for mistakes, we are talking sometimes one stroke of one letter, copied wrong, changing the word from men to donkeys, or things of that sort. Again, small errors that do not affect doctrine! Sometimes a difference might be the case of the word in Greek, or sometimes the tense of the verb - especially in Hebrew, where dropping a vowel point means a different vowel. (And yes, the vowel points were not in the original manuscripts, but that is because the people spoke Hebrew, and they didn't need them. When the Jews started losing Hebrew, the Masorets stepped in and added the vowel points for clarity and to preserve the unwritten aspect of the text)
The only reason I picked this quote from Wikipedia, is because it says exactly what my theological books say about infallibilty, so rather than type it all out, my fingers having some big issues these days, I copied and pasted. I would not normally use Wikipedia, unless it said exactly what was in my theology books.
I also compare between the modern translations and the Greek and Hebrew. It helps me to see the differences, but more important to appreciate the closeness of the texts. It says so much to me, that despite these human imperfections in transmitting the text, God totally preserves his Word. You don't need an inerrant text, you need to realize how God works in the lives of people to pass down his text, and for us, as long as 3500 years after some of the OT books, and 1900-1950 years later for the NT, we have the living Word of God to read, to study and to learn and grow from!
I hope that answers your questions. But feel free to ask more.
