How do you deliver bad news to your elderly mother? She "feels fine" although she coughs & is short of breath. She has stage 4 lung cancer.

This is the doctor's job, not yours.

I'd tell the onc that with this news, a 10-day scrip of Ativan (not Xanax) would help break it and provide her some clear, non-panicked thinking in what her next step should be. It's a legit concern. If it's hospice, then Ativan is part of the normal course, and in my opinion, it should be anyway.

Why on earth hasn't the doctor told her? You march her into his office and make him tell her while you and your sister are with her. He handles the medical questions, and you handle the emotional support.

That's how it went with my dad. He had NO idea he had cancer, and I went with him to his appointment to hear the results of an MRI from his doctor. He walked in and said, "Well, I'm sorry to say you have inoperable cancer in your liver. I'm afraid anything we could do for a younger person would kill you."

He then left us alone to process that for a few minutes, then he came back and answered Dad's questions. He sent Dad off to get an unnecessary X-ray so he could tell me that he had about a month, and that was that. We never saw the doctor again.

Your mom deserves to have her doctor tell her and to be there to answer her questions. You aren't qualified to answer them, and he should be there to do it.

I think any of us would be VERY emotional at being told we have but months to live. I think you are prepared for her to be, and she has a right to be, and you should expect her to be. I believe her doctor should deliver the news that this is stage four, and the prognosis, and should tell her that there is little option in terms of treatment, but that hospice can keep her comfortable. All forms that are not already done in terms of POLST and advanced directives need to be done asap.
The only thing you can do is let her cry, tell her that you will be there for her, and tell her that you will follow her directions, be certain she is medicated to keep her comfortable, get hospice care for her. You will have support of nursing, social workers and clergy if you wish it.
I am so dreadfully sorry. There is no happy way to receive this news. No matter how much drugging you do.
The doctors delivering this news will help with the denial but YOU must make it clear to him you believe she will go into denial so that he is HONEST. There is no good way for bad news. When I had breast cancer and said to my doctor "What do you think it is" He said "An occult breast cancer though it could be lymphoma". When I said "I never get sick; what can it be that is GOOD" he said "Cat scratch fever, but I don't think so". I could laugh about it later, but at the time it was shocking and brutal. There is no good way to hear very bad news. I just am so sorry.
As to the final way she reacts, that is something you have little control of. After getting the truth, you can live with how she chooses to handle it. If she never does accept it, then know that is one way to handle it and allow her the dignity of her reaction.

I'm sorry you have to deal with this, I think it's rotten of doctors to inform family members rather than break the news personally, face to face and available to answer questions. I think I would say something like "mom, the doctor called with the CT results and they say it's cancer", and beyond that take your cues from her. I wouldn't mention dying unless she does. And you don't really know at this point what the future will look like in the short term, so if she wants to go into denial and live as though everything is fine I can't see anything wrong with that so long as you have your plan B firmly in place.