Regarding the first and last parts of your question, if this is a valid promise, and if it’s quoted in the Shulchan Aruch, see YD 179:14 (h/t DoubleAA):
להשביע את החולה לשוב אליו לאחר מיתה להגיד לו את אשר ישאל אותו מותר:
To adjure a sick person to swear to return after [the sick person’s] death to tell him that which he asks [the sick person] is permissible.
Seemingly the oath would take effect, as it’s no different than anything else which isn’t tangible, regarding which the Shulchan Aruch rules in YD 239:3:
שבועה חלה על דבר שאין בו ממש כגון שנשבע שלא יישן או שלא ידבר:
An oath takes effect on anything intangible, such as swearing not to sleep or not to speak.
I don’t have a solid source regarding the middle part of your question, whether it’s permissible to recount the promise at the the burial. However, I suspect that one should be very cautious about doing so, based on YD 344:1:
מצוה גדולה להספיד על המת כראוי ומצותו שירים קולו לומר עליו דברים המשברים את הלב כדי להרבות בכיה ולהזכיר שבחו ואסור להפליג בשבחו יותר מדאי אלא מזכירין מדות טובות שבו ומוסיפין בהם קצת רק שלא יפליג ואם לא היו בו מדות טובות כלל לא יזכיר עליו וחכם וחסיד מזכירין להם חכמתם וחסידותם וכל המזכיר על מי שלא היה בו כלל או שמוסיף להפליג יותר מדאי על מה שהיה בו גורם רעה לעצמו ולמת:
It is a great mitzvah to eulogize the deceased properly. The mitzvah is to raise his voice, to say about him things which shatter the heart, in order to increase crying, and to recall his praise. It is forbidden to exaggerate his praise too much, more than is necessary; rather, one should mention some of his good attributes and add to them slightly, only that he not exaggerate. If he has no positive attributes, he should not mention about him. A sage or pious person - we recall for them their wisdom and piety. Anyone who recalls regarding one who does not have at all, or one who adds to exaggerate more than is necessary regarding him, he causes evil to himself and to the deceased.
Given how important it is that the eulogy be delivered properly, I would posit that, unless it’s absolutely necessary to bring out whatever point the eulogizer is making, one should avoid discussing it in his eulogy.