If I have multiple Gumshoos GX in play, can I use each of their "Search the Premises" Abilities or am I limited to only one total per turn?
Each Gumshoos GX can use its own ability once per turn. So if you have two of them you can use the first Search the Premises to have your opponent reveal their hand, then use a Trainer card such as N to have the opponent shuffle their hand into their deck and draw new cards, and then use the second Search the Premises to have your opponent reveal their hand again.
Source: Sun & Moon FAQ (2017-02-02), TPCi Rules Team (2017-02-02)