Yup... "I'll give you this chicken" would be correct. Should he complain as you're actually giving the chicken, about your general lack of generosity, you would then say, "Hey, I'm giving you this chicken." If you were transferring the chicken to him ceremonially (even if it were a moment before you gave him the chicken), only then might you say (with a formal, stain-glassed voice), "I give you this chicken as a sign that I will always provide food and take care of you."

Great question! We won't use present simple ("give") because it is a single action, not a repeated action or state. However, both future ("will give") and present simple ("am giving") are OK. "Will give" is much more common. The main reason is that using future tense allows a small amount of uncertainty. I believe that he wants the chicken, and will almost definitely accept it, but there's a chance that he might say "No, I don't want it." If I say, "I'm giving you the chicken," it is a statement of fact that cannot change. He cannot say "no." Therefore, it is a command. I am forcing him to accept the chicken.

In real life, a normal conversation will go like this. Big bro: Do you want my chicken? Little bro: Yes. Big bro while handing over the chicken: Here you go / Here you have it.