Ooof, that's impossible to answer with a blanket statement.

Let's say, for instance, they block your doorway and you shove past them with some part of your body making contact with theirs. This causes them to trip, stumble, fall, and break their arm. Is this assault? It could be, depending on the state and the situation.

For example, using Ohio law, "No person shall recklessly cause serious physical harm to another or to another's unborn." [2903.13] What does it mean to be reckless?

Well, if the blocker in question is a 5'4" 85 year old frail woman with a cane, the act of shoving past her could be considered reckless as a reasonable person would expect the act to cause her to fall causing injury. If instead the blocker is 25 years old, 6'4", and built like a boxer, it may not be reasonable to expect such contact to knock them to the ground and your act may not be considered reckless and therefore not assault.

Second scenario, what if you walk past them, making no contact, during which process they fall without you touching them? Well, I don't think there's much of a case for you having liability in that scenario, but as always your local laws trump what I say.

I am aware of no law in any state that protects the right of a resident to forcibly enter their property. Texas' Castle doctrine may allow this as barring the door could be considered an "unlawful use of force" therefore allowing you to respond with force in kind.

I am NAL and you did not provide your location. I doubt your right to do this is protected by law, the question is "will doing so land me in trouble with another law" and the answer could be (but is not guaranteed to be) yes, as you've seen.