While there is always risk in such things, if his professional credentials are good and the other comments don't reflect badly on that, I'd suggest asking for the recommendation and not otherwise mentioning it in cover letters and such. Treat it as a purely professional relationship, which it should be.
As you note it is odd to not have a recommendation from ...
If you don't get a letter from your advisor, and I certainly see why you'd feel uncomfortable doing so, the best thing to do would be to ask one of your other letter writers to address why you don't have a letter from your advisor. In practice, this may be tricky because it only works if you know that one of your letter writers knows about the problem with ...
I have written many letters of reference.
I have also been contacted, over the phone, by HR or admissions who wish to discuss the candidates, just to expand on experience or whatever.
So, if they wish to contact the people who you put as referees, then they will do so.
This isn't something you need to worry about. It is unlikely that they will do so, but possible. If they do, the language issue will not reflect badly on you. The people are peers and will understand that language can be an issue and will compensate for it. If your letter writers are positive about you in writing, it will be clear to the caller that they are ...
You have not stated where you are applying to. I can imagine that practices vary by geography.
However, in 25+ years of industry, academic, and related work, primarily in North America, I have found that it is exceedingly rare for hiring or admissions committees to reach out to providers of written references (this is in contrast to a habit in industry of ...
Say that you really wish you could accept her offer but, unfortunately, lack of money is becoming an issue for you. You need to start earning money (in order to live). With regret you will have to find a funded position even if it is not so desirable.
If he is still prepared to write you a reference then he will.
He can state that you were a student when he was teaching at X institution and that will be accepted.
If the admission committee feels it is necessary to contact him then they will do so.
I have sent reference letters to the Academic secretary and she printed them on letter headed paper - this ...
You are worried to much.
My PhD advisor also has several connections in HR. This can also be
damaging if a company background-checks through the university HR
First, HR is not allowed to say anything bad about you with no clear evidence (for fear of being sued). They will just give minimal information, and that's it.
Second, background check is ...
Currently, in the U.S., in mathematics, I think to contact some applicants' references but not others' would be viewed as inappropriate, unfair, etc. Applicants should be on a level playing field, with "level" in a very strong sense of fairness and equity.
You have to supply the contact information.
Telephone number(s) landline definitely, mobile not always
This is so that the admissions / adjudicating committee can get a reference according to their policy. That could be a 5 minute chat or asking the referee for a personal letter, or sending the referee a particular form to ...
Definitely contact them both. The email doesn't need to be the same to each, but inform each that you are also asking the other.
But try to write it in a way that gets you some feedback about what each would write in an actual recommendation: "I hope you view my contributions in a positive light. I found it a valuable experience...."
The senior ...
Short answer: The Ws aren't great, but they're basically unimportant in light of the rest of your portfolio.
I'm a little wary of withdrawals, because I know from personal experience that my own university's academic advising tends to use them as a way to help students avoid failing classes. So they're not great on your transcript, especially multiple Ws.