SATA drives typically work with SAS controllers. There are differences in the drive logic, and SAS drives are much better for servers. A SAS drive will most likely not fit into your coolermaster bay. If you take a close look at the connector, the SAS drive has a plastic piece bridging between the power and data cables. You would be able to plug in regular SATA power and data cables as that little piece of plastic will block them.
I have to admit I chuckled a bit when I saw the picture you posted. If you had a regular caddy in your hand you would have figured out that was just a blank very quickly
2.5" drives are actually a lot more common for servers than you might think, for a number of reasons. Having the smaller diameter platter is beneficial as the surface speed at the head remains more constant. There is also generally less vibration with the smaller drives as the platter weighs less and is smaller diameter. Also, you can simply fit more 2.5" drives in the same space as the 3.5" drives. In a desktop, who cares, but in rack mounted server land, it makes a big difference.
Last 350 series I had was a rack mount, and didn't have those big trays. I'm guessing those are for cabled 3.5" drives or optical drives.
Used hard drives are definitely a no no, but to be honest so is a used server! You might be best off getting good condition working pull SAS drives from eBay. Get the HP drives. They have a green and amber light. Set up RAID for redundancy. If a drive fails or is in predicted failure state the green light goes out and the amber light goes on. Let your client know to keep an eye out for it.
You are learning here. At the end of the day you'll have a dated but high quality server. You will be able to replace parts as they fail without complete system failure. Eventually your client can get a new replacement server and the transition will not be hard. Then you can learn how to migrate a server.