Do you have a nickname for your pet?

Yes, but it is private between me and her. Besides it is against the TOS.

I call my small cat "kitten-sez".  At 9 years old, he barely weighs 6 pounds.  compared to my other 14 pound "mini panther", he's like a klitten.

We have two rescue dogs, both female.  Each has a name; but when I am being affectionate with either, as opposed to feeding, medicating, playing with either, etc., I tend to address each as "Little Girl.)

Cat #1 is named 'The Mute'. but he is a loud vocal beast who is actually deaf.  Cat #2 showed up in January just as the really long cold days of northern winter arrived. He is about a year old, very athletic and likes to jump up and catch birds and butterflies. I named him 'Rob' after a top ranked motorcycle racer who also rescued a homeless kitten last winter. 

We have a cat named "Spook." I sometimes call him variants of it, Like Spookie-ookie, Spookie wookie, and Wookiee, which fits because he's very loud and communicative and sounds like one. 

My kids also took to calling my old dog "Grandma" because she was old as well as very gentle and loving. She actually answered to it. She passed away about a month ago though. :/ 

Sometimes yes and sometimes no, depends a on the Pet and the connection. Our old cat took on the name "killer". 

Graduate Student housing anywhere is mouse invested as a given.  "September" came home as a kitten.  She proved so adept at ridding our unit and other Grad Students units that we all gave her the name "killer Luke", aka Killer. 

Our cat, Suri, is often called Madam because we respect her right to rule the household.
She's Coquette in her affectionate, smoochie moods, Kitten or Tigga in her playful moods.
She gets lots of other nicknames, new ones every day.

We just have a name for her, but why bother? She doesn't come when you call her.

She is quite voiceterous. (A word made up by my wife. I like it.) Training her? She has us quite well trained.

Yes, yes I do.