That would seem to indicate that everyone is considered Gentile (as there is no distinction and all are the same).
In a sense, yes. Nationally, all nations are now as all nations were at Babel.
But Paul seems to preach that Israel is not cast away (Rom 1:1-2).
Right. When he wrote that, they weren't yet. Stumbled, fallen, partly blinded in unbelief, but not discarded. I think of it like this: that branch was broken off, sure enough, but it was still in God's hand. He hadn't cast it aside yet, so He could have still grafted it back in (Rom 11:23), pending their repentance. But we know they didn't. That's the only way it makes sense to me.
And that there is still a distinction between Gentiles and Israel (Rom 11:28).
Yes, then. Jews were even then personally no better than Gentile dogs (Rom 3:9) but God was still appealing to Israel.
There is no distinction for the BOC.
But I'm not sure we can flatly say that there is no Israel (distinct from Gentiles) at all at this time.
I'm always willing to be shown wrong but a lot of Romans makes no sense to me if God was not yet, at the time Paul wrote it, still in some sense dealing with Israel as Israel.