why should gravity still affect them?
Why not? Interaction is selective in the real world, too: some matter is affected by electromagnetism, some is not.
Backstage technobabble speaks of shields being a gravitic technology. That would jibe well with this phasing stuff, because these gravitic shields apparently easily stop matter phased by the transporter from getting through. OTOH, phasing comes in degrees, and enough of it will apparently allow the matter to penetrate even intense gravitics...
why should the suits the landing party was wearing not start shifting through their own bodies?
Or their left arms through their right lungs?
It would make sense that matter that has been recently brought into the field is more solid than matter that has spent time in it. Hence it, and all
of it, goes through the older matter if sufficient force is used. It doesn't matter whether the new matter is people, or spacesuits, or dust, or air.
Of course, the phasing done by the transporter merely appears to take an entity from this world and move it intact into a parallel realm; the "phased matter" isn't phased to itself, it's only phased vis-á-vis the normal, unphased matter around that wasn't subjected to the transporter. The phase cloak thing no doubt works the same way: LaForge doesn't become transparent/penetrable to LaForge, merely to the rest of the universe. All the forces of nature that normally keep LaForge together still keep on working, only now in their phased form.