In all, 10 players with Kentucky ties were selected in the 2016 MLB Draft: 4 current Cats and 6 high school recruits.
Below are the results from the draft, along with the signing bonus pool amount assigned to each pick. Note that these amounts are not guaranteed - teams can go above or below the assigned number (subject to some constraints and other rules), but it will give you a rough idea of how much money each player is likely to be offered to sign.
Everyone went pretty much where they were expected to. I'm particularly excited for Dustin Beggs going in the 16th round - last year he was taken in the 32nd despite dominating the SEC. While the Marlins are always going to be dumpster fire as long as Jeffrey Loria is the owner, it's great to see Beggs rewarded for his two terrific years in Lexington.
Who's Signing and Who's Staying?
This is pretty easy - everyone drafted off the team is going to sign unless something really bizarre happens, and only Brown and Shelby even have the option to return to school. I expect the majority of the high school players are also going to sign (again, barring something odd happening) with the exception of T.J. Collett and possibly J.C. Flowers.
Collett will most likely play college ball for three years, though whether that's at UK or another school remains to be seen. Flowers - based solely on his draft position - is more of a question. If you look through Baseball America's Draft Database you'll see that a lot of high school players taken in the 18th round do end up signing, but there are enough that don't that it's not automatic. However, given UK's current lack of a head coach my guess is that Flowers ends up signing with the Reds.
Given the draft results, I'd say it's actually a little bit less imperative that Mitch Barnhart fill the head coach position soon. Realistically, the only player to whom it might make a difference is J.C. Flowers. Brown and Shelby are going to sign regardless, ditto all the HS players drafted before Flowers. Collett will probably choose school, so the only work the new guy will need to do is convince him to stick with UK instead of heading somewhere else.