'bien' restricts the meaning of 'aimer' to 'like' ... so j'aime => either "I like" or "I love" ... but j'aime bien => only "I like".

"Bien" is an optional word here. It is not nor has it been required with aimer to mean "like" in this context.

Both J'aime bien promener mon chien and J'aime promener mon chien are acceptable. J'aime bien is closer to 'I quite like walking my dog' so it's definitely like not love. J'aime on its own could be read as I like walking my dog or I love walking my dog.

It's not.

It is correct, but I don't know if Duo will take it or stick with promener (for the sake of teaching you that particular verb).

I think it actually sounds more natural. As a French person, I never say "promener mon chien". Nor have I ever heard anyone say that.

"Sortir" = to go out, to take out. So if you take out the talk, it would probably be understood that you are facilitating the dog's business. In this sense, the term "take the dog for a walk" could be understood as a euphemism for the same thing. But "sortir le chien" would nevertheless be translated as "take the dog out". If does not mean you are necessarily going for a walk, e.g., to exercise the dog.

You need a "se" when you take a walk yourself (se promener). Honestly, you almost never use the verb "promener" (without se) except when followed by a complement (mon chien, for example).

marcher describes the action that is putting one foot in front of the other. But when talking about going for a walk or taking a walk, or walking a dog promenade is used.

Because whether a 'de' or 'à' or nothing is used before the infinitive depends on the verb before the infinitive and aimer does not have a 'de' after, it just is followed by the infinitive. J'aime faire quelque chose. http://www.collinsdictionary.com/dictionary/french-english/aimer

Walking your dog is a walk with a purpose. Balader as I understand it is more of a wander with someone or something, meandering. But there are some google hits for it so you could suggest it as an option. Especially for those situations where it's not that you are walking the dog for your dog's benefit, but for where you are going for a wander around just with your dog too.

Promener is walking something or someone, where it is implied that one is leading the other (a baby for example - je promène mon bébé). Se promener is when people walk about (no one is leading the other....). Je me promène - I am taking a walk or I am going for a walk.

It just is not. Promener could be better translated as strolling and it is the term used for walking pets in French.

You walk yourself (je marche) but you « promène » your dog.

Promener has a connotation of a leisurely walk. You use promener for pets/animals or even babies. You use se promener when you are taking a leisurely walk with someone (je me promène avec mon ami).

You only use promener when you walk a pet.

No you only use promener to say you walk a pet. You could say je marche AVEC mon chien (I walk with my dogs) but the meaning is a bit different.