This has probably been asked before, but I couldn't see anything with a quick search.

What's the timing of Deadman's Switch and an emergency deployment?

Say I'm using Bossk in the Hound's Tooth with Nashtah Pup in reserve. Bossk is destroyed.

Both Deadman's Switch and the rules for emergency deployment on pg 10 of the Rules Reference say "after you are destroyed". Specifically, emergency deployment says "after a carrier ship is destroyed, but before it is removed from play".

As the timing of these two effects appears to be the same, am I correct in thinking the player gets to choose the order they happen in?

Can you trigger Deadman's Switch to damage any ship at range 0-1 and then deploy the Nashtah Pup before removing Hound's Tooth, so as to avoid also damaging Nashtah Pup if you only wanted to perform a 1 speed maneuver to deploy?

Yes, as you choose which order to resolve the ability queue.