My only experience with an LNA is from when I bought a cheap one off of ebay and killed it by changing antennas while powered on. 
Is the new FA Pro Stick susceptible to this sort of damage? Should it be powered down before cable changes, or is it tough enough that it does not matter?
It is static sensitive like most other electronics, though I think thereās a protection diode which mitigates it somewhat.
The dongle antenna port isnāt sending power towards the antenna or anything like that so a shorted or open antenna shouldnāt break anything.
Ground yourself before touching the connector and you should be OK. I swap connectors āliveā regularly and I havenāt broken anything yet 
If the dongle is directly connected (no USB cable) you may have mechanical issues, you donāt want the USB connection wobbling while you unscrew the antenna connection.
Oh, so THATS how I killed my cheap ebay LNA.
(at least they refunded me)
But yes, I too have switched antennas, cables, etc with the prostick powered on and connected without any issue.
A filter, like the superb FlightAware filter, will provide some protection against static and other abuses, even though itās not primarily designed as a protective device.
If youāre not using a filter, a common approach in industry is to put an attenuator in line, such as a 1dB attenuator. Yes, this introduces signal loss. It also introduces a DC path to ground that will protect you from static charges and other nasties. You can usually find 1dB SMA attenuators on eBay for less than the cost of a new SDR.
bob k6rtm