If I am using my Gmail account overseas. What date and time shows up on the recipients email in the U.S.? Is it my date and time overseas or the recipients date and time in the U.S.?
As far as I'm aware, your client sends the mail with a timestamp containing both time and zone; then the receiving client (such as gmail) may adjust this to the recipient's zone, matching their local clock. However, as in "falsehoods programmers believe about time", there might be hiccups, such as outlook's messages from the future (to an extent I've noticed this on gmail, in minutes), and thunderbird doesn't connect messages; when you have a reply chain, might see a plaintext view of the originally received header (including their interface language, people's nicknames and timestamp).
Tldr if both correspondents are on gmail, each sees their own time only.