1 A. They won't appreciate us. [Correct. In the past, present and future forms, the auxiliary verb - will (not) - indicates tense and the verb (in this case - appreciate) remains in the infinitive form.] 1 B. They won't appreciated us. 2 A. I am afraid of dogs? [Correct. The preposition 'of' means the fear is about dogs. When you use 'with' it means you and dogs are afraid together presumably about something else.] 2 B. I am afraid with dogs?

Your question should have been "Which ones are correct?"

"True/false" refers to reality. If something is "true", then there is proof of it in reality. You meant to use "correct/incorrect". If something is correct, it is technically perfect and without errors. 1. "They won't appreciate us." If you use "appreciated", then you need "to be" in front of it: "They won't be appreciated (by us)." However, the action is completely reversed. 2 "I am afraid of dogs." This is a collocation you just have to learn: afraid of ___. No other preposition fits.