All dogs vocalize. It's a species thing, not a breed thing. So it is perfectly normal for a dog to use his voice to communicate. The difference between "yappy" and communicating is two fold. The dogs accused of being "yappy", often have high pitched, annoying voices. (not really their fault!
) We are lucky there, because, for their size, Havanese tend to have fairly low-pitched barks, like a bigger dog. (though your little puppy is probably still pretty high pitched, just because he's a puppy!) The other reason for calling a dog yappy is because he uses his voice inappropriately, often barking without apparent reason, and not willing to stop. Some breeds are more prone to this than others, and some individuals are more prone than others. As a breed, Havanese are not big barkers, (in comparison to other small dog breeds) but there ARE individuals who bark more than others.
One thing that can be VERY useful, and your little guy is JUST the right age to start this, before he develops bad habits, is to teach him to "speak" (bark) on cue. This may seem counter-intuitive, but it is often easiest to teach behaviors in pairs, so you have an on/off switch. So once he understands to "speak" on cue, you also start alternating with "quiet" on cue. Then you have a built-in "off" switch!
As far as him barking at the other barking dog, this is something that will be very hard to teach him not to do. Two confined dogs are hurling insults at each other.
Your best bet is to bring him inside, where at least he can't see the other dog, and then distract him with a training session, to take his mind off the other dog, and tire his brain out. That way you can be rewarding him for something positive, (paying attention to you and doing what you ask) while preventing the unwanted behavior. It is MUCH easier to teach them to DO something than to "not" do something!
It sounds like the other barking he is doing is largely "alarm" barking, where he is barking at something that is (to him) novel in his environment. In those cases, try to calm him down by getting him to approach the item. (don't force him, but offer treats for getting closer) Once he has checked it out, he will probably stop barking on his own. This same idea works with people at the door. I have NO problem with my dog alarm barking at someone at the door, but I want him to stop once I have answered the door and greeted the person as an "OK" person. So when he was a puppy, I always had cookies in my pocket. When I answered the door, I would give a few cookies to the visitor, and ask them if they would help me train my puppy, by squatting down (you don't want them looming over the puppy, because THAT can be scary!) and giving him a couple of cookies. Before long, he realized that when I said "Thank you, Kodi, I've got it!" that this was a "friend". He now settles immediately for a known person, and very quickly if I acknowledge the person as "OK".
Finally, it is pretty common for puppies that are VERY quiet when you first get them to suddenly "find their voice", and be rather noisy for a while. It is a normal part of growing up. It is how you handle it that will determine what happens later. Help him learn to use his voice appropriately, to warn you of perceived danger and tell you when the water bowl needs filling. Teach him that it's NOT such a good idea to be swearing at the neighbor dog!