Even if your gauge is faulty, max pressure ratings have a big safety margin, so a gauge error of even 25-30% wouldn't be an issue. In any case, over-inflation is a fairly instantaneous event.

I suspect it was a tire seating issue, so a question. Has this tire been on a long time and were you just topping off. Or it is it a recently installed tire, or a flat repair, or has the tire been empty sitting on the floor and you were pumping from zero. The odds of poor seating being the issue depend on your answer, since a tire that's been OK for a long time wouldn't suddenly get unseated.

As for the tube, odds are it'll be OK, since tubes are only balloons and have decent ability to stretch quite a bit (usually to double rated size). It might be more prone to splitting if punctured, but should be OK until then. The biggest issue with undersized tubes is that they're stretched thinner, and so the natural pressure loss that tires all undergo will be faster, so you'll need to top off more often.

When you install the tube, pump to enough pressure for he tire to take shape (15-25psi), then spin the wheel and make sure the tire is seated evenly all the way around by looking at the molded line just above the lip of the rim. You can sometimes also detect a poorly seated tire by wobble.