Strange ruling, so essentially they are saying that all groups have to be open to anybody to receive funding. So the Black Student Union couldn't turn away members of Klan. The College Republicans couldn't turn away Communists and vice versa. It seems to open the door to a lot of shenanigans, if a group opposed to a groups ideology decides to "take over" that group. Of course an easy way around it is to simply not apply for Government funding.
Bingo.
First, the answer to all your hypotheticals is "yes". Second, and this is what is a little misleading about this particular case, the Christian group didn't really receive public "funding". They were, like any recognized group on campus, assigned a building and room in which to meet. That, say the justices, constitutes "public funding"; access to public property.
I haven't read the decision in it's entirety yet (I'm barely a 1/4 of the way through MacDonald), but I'm interested especially in Alito's lengthy dissent. I'm worried what this might do to the Boy Scouts of America, who receive no public funding but are allowed to use public property.
They are not saying that every University has to have that policy. In this case, the University claimed that that really was their policy, and the Court held that the policy was not unconstitutional if it was applied to every group and not just used against the Christian group. The case was sent back to the lower court for hearings on that issue.
This has been Hillsdale College's core principle from the start, in order to avoid any governmental regulation and imposed value system onto their excellent conservative educational system. No student grants or subsidized loans are allowed. Still, the college was sued and forced to allow some of this government funding to seep in, destoying in some measure their right to autonomy.
Expect this for groups which shun government funding. They will be forced to submit by other means. We no longer live in a free society.
A "hostile merger" in the marketplace of ideas.