What is the most appropriate phrase for touching i.e. a dog or a horse in a gentle manner? My first guesses are 'to pet' or 'to fondle' or 'to scratch'. Anyway, the other meanings of those words let me conclude, that there need to be more appropriate words.
You can definitely pet or stroke your dog.
Though pet may have other connotations if applied to humans, it's perfectly acceptable to pet your pet.
Scratch is OK.
If you're going to scratch your dog's head, people are going to generally take that to being a gentle action - they really wouldn't assume without other context that you were about to attack him with barbed wire ;)
There's a diminutive form of scratch you could also use - scritch. You're probably not going to find it in any dictionary, though I have a vague idea it may have originated in the Peanuts [Snoopy] comic strip.
You can also pat a dog or horse, though that diminishes with size. You wouldn't want to pat the cat, she wouldn't like it.
Fondle is definitely to be avoided.
That is really never going to give the right impression. It has only sexual overtones & no 'clean' alternative definition.
No results in OALD but I found it in Oxford Living Dictionaries
US [with object] To scratch; (now) especially to scratch (a pet) affectionately with one's fingernails.