You can accept an offer and negotiate the start date at the same time. Accept unconditionally first, then raise the date as a request with your reason — usually a notice period — so acceptance is not read as conditional.
Most employers ask for a signed offer letter, but a written acceptance email alongside it creates a clear timestamped record of the terms as you understood them.
Yes. It costs one sentence and catches discrepancies between what was discussed and what was documented, while the offer is still easy to correct.
Practically, no. Negotiate before you accept — once you have accepted in writing your leverage is largely gone.