Sorry you are getting hassled but they mean well. But you didn’t ask for help to figure out if you are being cheated on.
So, you can simply take a trip with her out of the country. Go on a cruise out of Singapore for example, or to visit Universal Studios, essentially doing a border run together. Thus way, no visa. Maybe visit an agent next time to ask for advice on how to proceed given you plan to marry. Good luck!
Yes, you can overstay a few days without much concern! I would rather pay a 2 or 3 day fine than visit immigration or make a border run. One would not make a habit of overstaying, but you realize that.
Been there plenty of times as single male and had no issues. The immediate. The immigration line can be long. If your passport is in order and you are pleasant and respectful, you should be fine. Of course, you know not to bring in anything illegal, but I’ve never had my bags either coming in or leaving inspected. Relax.