I will say, regarding my previous responses, and you probably are already more tactful than this without having to be told - unless the physician has already told the patient they are dying and/or the patient has used that language themselves, I would probably hold off on using that particular phrasing when you are talking with patients. 1) obviously you want to be sure it is accurate lol, and 2) that can be pretty jarring for a patient to hear for the first time, and I think it is best to come from the physician in the context of a larger conversation.
If the patient and family keeps refusing, I respect their wishes. I may continue the conversation or revisit goals of care if things change in the future, but patients have the right to make their own decisions even if I disagree with them. Even under the best of circumstances sometimes people need time to come to their own conclusions and be ready to take the next steps, or they might never want to transition to hospice because of their goals, values, preferences, etc. And that's okay - my job as their family doctor is not to force them into what I think is right, it's to give them the information and support they need to make the right choices for themselves.
Regarding the second question - the physician can choose to NOT offer treatments or interventions they feel will be nonbeneficial and potentially harmful. e.g. an oncologist might say a patient is too sick for another round of chemo, a nephrologist might say a patient is not a candidate for dialysis, a surgeon might decline to operate, a GI doc might not offer a long-term feeding tube. The family/patient can refuse any type of treatment, including hospice care, and the physician doing that care anyway would essentially be assault. There may be extreme circumstances on the level of neglect or abuse where there may be a need to seek a surrogate decision maker for a patient but this is quite rare.