Does lowpass filtering and decimating improve SNR?
So, let's not forget what SNR is: it's a relation of powers present.

The thing that improves SNR is a propoer low-pass: it leaves the signal power alone and reduces the power of the noise.

An ideal low-pass filter will leave zero noise outside its specified bandwidth – so it doesn't matter whether you "cut off" these bandwidths using decimation or not, from a pure SNR perspective.


  This question is of practical importance: when measuring a phenomenon, say at 1Hz, is it useful to use a very high sampling frequency such as 1kHz or more in order to improve the SNR by filtering and decimating (given that the memory and computational cost is not a problem, of course)?


Yes! That is called oversampling, and it's useful for a couple of reasons, foremost for doing so-called noise shaping, which allows you to get a lower noise floor in your band of interest. With appropriate filtering, you do get a lot of SNR gain.

This is actually done pretty often; Software Defined Radio devices often run at dozens to hundreds of Megasamples, no matter how much bandwidth you want to observe, and do the filtering and decimation in-device. That way, you get the oversampling benefits, and also, you don't waste money and potential phase linearity etc by implementing steep-edged analog anti-aliasing filters, but can get away with much more relaxed filter specs.