What sampling frequency do I need for a non-periodic signal?
You have a screen shot - why don't you plug a memory stick into the oscilloscope and take the readings into excel. The data will be high resolution of course but in excel you can average, decimate, filter, decimate and average or do whatever you feel fit to reduce the data until you are starting to lose temporal and amplitude information that is important to the quality of the parameters you wish to know.

When you reach that point you will know what sampling rate is sufficient.

As for PGAs inside a chip with a combined ADC - you can have them separate but still controllable digitally. I use a PGA from Microchip for this - the MCP6S21-I/MS and we measure strain gauges and load cells.