The letter panel looks like a quest board and is not one. Nothing in it is tracked, nothing appears in the quest log, and no letter ever schedules anything at the other person's end.
Asking to meet (会面) does not ask anyone anything. MailView.OnClickSendVisit writes your letter and
then immediately composes the reply itself:
There is no meeting, no marker, and nothing to go and do. The brush-off carries no cooldown: send again later and you get the real answer if that character is active by then.
The 索要礼物 letter with an 接受 button is settled in the panel, not in the world:
It never becomes a tracked task, so there is nothing to look for in the quest log. Come back and press accept when you have the item.
It pays no relationship. The accept path removes the items and posts letters and does not touch 好感 or 情分. Handing the same item over in person does: the gift panel converts an item into both, scaled by its quality, its level and that character's taste for it, and shows you the numbers before you confirm. Mailing away an item that could have been a gift is throwing it away.
MailManager.SetMailFinish fires a "letter finished" trigger, and the letter-condition table can hang
new letters off that, so closing letters advances the correspondence. Nothing in the game shows that
it does.
When a gift letter (送礼) arrives, its items are created directly in your inventory with no collect button and nothing to click. The letter is the receipt. If someone "owes you a drink", check your bag rather than waiting for a prompt, though the promise itself is flavour text drawn from a random pool, not a debt the game records.