Sampling below Nyqust Frequency
One does need to (not a necessary condition)! But if you do, you can recover (at least theoretically) the continuous signal from the discrete samples (sufficient condition).

The bandwidth most usually  denotes the span between the zeroth and the maximum frequency (in the analog signal). Sometimes, for a band-pass signal,  it denotes the span between the minimum (positive) and the maximum frequency, and there are similar theorems: twice the bandwidth (sometimes a little more depending on the location of the minimum or the maximum) is enough.

If you sample below, aliasing might occur, but not every-time, this depends on the structure of the spectrum.

You care only about 1 Hz, it is dangerous to simply resample: it is advised to filter a little above 1 Hz (say 1.5) and then resample consistently with 1.5 and the precision of your filter. Could be 4 Hz (above $2*1.5$ Hz). Those figures should not be taken for granted, just an idea of the whole method.

Why what it set to 100 kHz initially? Several possibilities, for instance:


people did not know initially in which part of the spectrum the information was located.
the acquisition scheme  was designed in a "the more data you have/the more information" mood