I'm pretty much like this. I only get emotional about things that are rationally meaningful, about things that make sense. As a result I can turn my emotions on and off at will. Though they are on by default. If something is meaningful then I may cry. But for example if crying at it would be inappropriate in some situation that I may be in, then I will not cry, because the meaningfulness of acting charitably in that situation overrides the meaningfulness reacting to the object.
I don't have a pet, but in my estimation I would not cry when it dies. Simply because I already know that things die, so I don't get any new, meaningful information when the death does happen. Furthermore I think I could also shoot my dog, if it is necessary. Simply because the meaningfulness of the act is that it's objectively morally good, as proven by the words 'if necessary'. I wouldn't be sad over an act that is objectively good. Shooting your own dog is objectively more loving FOR THE DOG (love is cosmic yo) than letting him live. So why would I be sad over an act of love? I MAY get sad over some new, meaningful information that may come by the fact that I had to shoot my dog instead of the dog dying of natural causes. This is of course hypothetical, but I just extrapolated what could happen based on my current idiosyncrasies.