Mostly when I see other animals they wander or run off through the bush as soon as they see me. I have had one of my horses decide she wanted to follow the moose through the bush. So, I let her. We had a jolly good time, but after 45 minutes I steered her back onto the trail.
I have had dogs chase me. I turn my horse around and chase the dog! :) What fun! Can you tell I like to chase animals? I once had a german shepherd that wasn't scared of my horse, so I went up beside him and kicked him in the head -- just took my foot out of the saddle and kicked him. That sure surprised him. Just make sure you have a really good seat.
Bears -- mostly they will walk away. If one starts toward you, I would try to walk away from it at an angle -- not my back turned, just as though you were going another way even if the bear wasn't there. If that doesn't work, make a noise. The bear will likely look up as though to say, "Oh, what are you doing here? I guess I'll go elsewhere." Deer never charge. At least I've never heard of it. Moose will charge generally only if in rut and you are in their area. Or possibly if your mare is in heat. You can't outrun a moose. I've never had a moose charge me, he will warn first by shaking his head. At that point, walk away. He will see you leaving and be ok with it.
Coyotes I have never seen while on horseback except waaaay over there in the field. Fischers come out only at night. Bobcats won't go for a horse. Mountain lions -- ya well.. not sure what to tell you there. Like you say, they are rarely seen. We do have them, I have heard them, I have seen their prints, but they are so rare that the provincial wildlife ministry only FINALLY ADMITTED that they are in Ontario this past year. After MUCH public outcry. Anyway, if you hear of any in the area you are riding, stay away from rock-cuts and tall trees. Lions like to sit higher than their prey.
Of course, there are ALWAYS exceptions to these things, but I have spent 9/10th of my life living in rural Ontario and have never ever been threatened by a wild animal unless I cornered it. (dumb kid, dumb kid...)
I don't know what the gun laws are in NH. Here in Ontario they suck and I can't carry while riding, but lots of people in different areas of the states do. I would if I could, though you'd definitely have to practice on your horse.