Can a software defined radio with a frequency range of 300 MHz – 3.8 GHz detect AM (535 to 1705 KHz) and FM signals (88 to 108 MHz)?
For those given specs, without additional hardware (upconverters) or modifications to the SDR circuit, it's not possible to detect AM or FM (88-108 MHz) range of frequencies. As you might have also seen, there are frequency range extenders (especially on the low frequency range) available as kits.

Also, the sampling rate determines the maximum bandwidth of simultaneous detection (or spectrum analysis if you prefer to use the SDR for that purpose). So, the bigger the sampling rate the larger will be the spectral bandwidth that you can capture instantaneously. This means, more radio channels are observed at once and more carriers are searched without changing the central tuning frequency of the SDR tuner chip. Which makes it faster in spectrum scanning applications. 

However, using a large bandwidth (high sampling rate) mode of SDR, might bring the problem of transmitting the data from SDR to PC, especially on a USB 2.0 interface. (this is not a problem on most ubiquitous SDR receivers used at 2.4 MSPS rates) Also more processing and interrupting than necessary (provided a narrow bandwidth radio channel is to be demodulated) is done at those high sampling rates, to decode just a small portion of the analysis bandwidth.