In short, yes, they did.

The Order of St John of Jerusalem based in Malta had numerous ships sailing under its flag and corsairing was extremely frequent and a very profitable activity. They would target rich Muslim vessels and take the booty onboard - this being cargo, currency, or people. It got to the point where the Venetians actually held the Order responsible for damages through the means of a "sequestro", in which they would hold their landed estates to "ransom" until they stopped plundering so, uh, liberally. The Venetians were caught up in this, as the Ottomans traded with them and thus considered them responsible for the damages incurred.

For example, in the book "Knights, Corsairs, and Slaves in Malta: An eyewitness account" by Thomas Freller he recounts the slave market in St George's Square in Valletta:

"Large fortunes were made by corsairs by selling captured Muslims on the slave market in Malta. Like the ships, the cargoes and the captured men and women were also sold by auction to private citizens, merchants, and even the Order's treasury. It was indeed common knowledge that the Order was one of the big buyers of galley slaves. [...] The slave market was held in Piazza San Giorgio opposite the grand master's palace. The prices for single slaves differed from 30 to 300 crowns, as another seventeeth-century traveller observed.

Other good sources on this particular subject would be "Venice and Hospitaller Malta 1530-1789: Aspects of a relationship" by Victor Mallia-Milanes, and "Hospitaller Malta 1530-1789: Studies on Early Modern Malta and the Order of St John of Jerusalem" by the same author. Also useful is this webpage, if not for nothing but the references: https://vassallohistory.wordpress.com/corsairing-activities-the-order-of-st-john-and-malta/

With regards to other places in Europe (or at least Christendom as the question relates to Muslim slaves) I can't really tell you anything, but hopefully somebody else can. :)

Yes, definitely. Every Mediterranean power during the 1500's and 1600's depended on galley slaves to row their vessels. Some of them were domestic prisoners, but at least during the 1500's, the majority were captured from foreign powers.

So, European powers from Spain to France to Italy to the Knights of St. John would routinely capture Muslim slaves for their naval forces. (The Ottomans were no better, except that many of their galley slaves were Christians from their own empire.) And, as their slaves died of harsh conditions in the galleys, they'd need to raid for more slaves in a vicious cycle.

For a very accessible survey of sixteenth-century Mediterranean warfare, with details on how the galleys were manned, I'd recommend Roger Crowley's The Battle for the Center of the World. Or, W. H. Lewis's essay "The Galleys of France" has a view of French galleys in the later part of this period taken from a Huguenot prisoner's account. (W. H. Lewis was the brother of famous author C. S. Lewis; you can find his essay in the Inklings' collection Essays Presented to Charles Williams)