I love that every Sci fi show eventually does the "out of phase" episode, and I love seeing the fun twists and flavorings they use to make it "their" version.

Like Stargate sg1 and their crystal skull radiation and advanced dimension-shifting technology episodes.

And my personal favorite lampshading for this is actually from Stargate. in the in-universe parody show "Wormhole X-treme" one of the main character parodies asks "if I'm out of phase, why don't I fall through the floor?" and the response from the writer/director/producer/whatever he was, was something along the lines of "I'll have to get back to you on that"

Multiphasic structural integrity fields

I believe there was originally a line of dialogue about there being resistance when they pushed through objects, which was supposed to explain why they don't just fall. That line was cut, but IIRC it is still referenced indirectly when Geordi's hand gets zapped and he says that there was more resistance to pushing through stuff afterwards.

That wouldn't explain why there is enough resistance in the floor to stand and walk, but not enough resistance in walls to prevent them from easily passing through. Presumably their mass and the pull of gravity are unchanged, so the resistance would have to be enough to counteract their weight. And even if they did weigh less, they still propel themselves forward through walls by pushing off the floor, so either the floor needs to be more solid, or they should be nearly weightless and move by paddling their feet through the floor until they build up momentum enough to smash through a wall. Also, if they are applying pressure to objects they pass through, shouldn't people they touch feel it?

Personally, I'd probably explain the floors specifically being impassable by blaming it on the way the artificial gravity is generated.

I don't have a good explanation for how they can breathe, how they see without interacting with light, how they can hear clearly when matter isn't really touching them and therefore can't conduct sound, etc.

And how did they breathe?

I looked it up on Google, they couldn't "look down" and thus couldn't fall.

Don't even ask the harder question of why they were able to stay inside the ship as it moved.

I like to think that they did... a few inches... till their shoes hit the spinny-ma-thing gravity generator below the deck.

Obviously, it has to do with basic operations of the main deflector dish polarising the outer hull and parts of the floors

Question for the writers or Q, not Picard.

Another question: How you fall down in space?