Are you able to do anything other than beat up and or tranquilize your troops while on mother base? Some of my troops ask me to train with them but it seems like all I am able to do is attack them. Additionally, whenever I do attack them, they thank me for it and I can't tell if they're being sarcastic or not. As a follow up, am I able to pet my puppy while on base? I feel kind of bad about not interacting with it while it is barking at me.
To answer your title - No. I've heard a theory that the reason Big Boss doesn't salute back is because the last person he saluted was The Boss (maybe also why he refuses to salute in the ending cinematic of MGS3)
For the other parts of your question, Punching and tranqing them is all you can do with them - Them thanking you is sincere, it's sparring with them essentially.
And finally, petting DD is something I believe that can only be done out on the field.