Is it invalid to say that some event might have "happened" for some observers in the universe, but not for others?

None of your questions are invalid. While it's true that there is no absolute definition for the concept of "now", that does not make it an invalid concept.

When you ask "What happens in Andromeda now?", most people would assume you are asking what events are occurring in Andromeda at a time that would be described within your own reference frame as simultaneous to the current moment. The obvious answer to the question is "I don't know because the information of those events hasn't had time to reach me yet", but the question still has some validity and can eventually be answered retroactively.

That said, since Andromeda "now" is space-like separated from here and "now", there are observers in other frames of reference that would (eventually) disagree with us about what is happening now in Andromeda. For some observers, it's entirely possible that the events we would recognize as having happened "now" they would say were yet to have occurred. So, in a way, you could say that some events haven't yet happened for some observers in the universe. Or rather, you could say that there are some events that we would say happened before now and some observers would say happened after now.

It should be noted that if two events are time-like separated, then that means there exists an absolute chronological ordering between them. Essentially, what that last statement boils down to is that if we are currently aware of an event that happened before "now", then there does not exist any observer anywhere that would claim it happened after what we call "now". The only events for which there could be a disagreement about whether or not they happened by the time we reached this exact moment of "now" are those that we are too far away from to know about yet.

Of course, just because there is a disagreement sometimes about chronological ordering, it doesn't mean the notion of asking what is happening "now" is invalid. It's just a bit ambiguous and partially unhelpful.

As for asking about what happened at the big bang: The big bang is a moment of time, which means if you point to a position in space that we'd recognize as being at a later point in time, everyone else would agree that it's a later point in time. Additionally, since the big bang represents a moment of time defined as the time when the universe held a certain property, all observers would say that what happened at the big bang was that the universe held that certain property. It's really kinda trivial.

In Special Relativity, every event “happens” for every observer. But different observers will assign different coordinates to them and they might disagree on whether one event is before, after, or simultaneous with another event.

In General Relativity, “horizons” can prevent events from being seen by all observers.

Your question talks about what seem to me to be two different notions: "happened" and "happened now."

What relativists usually talk about is whether event A is in the past light cone of event B. If event B is on the world-line of an observer, and A is inside B's past light cone, then A is an event that, at least in principle, could have been known to the observer at B.

Relativists don't normally talk about "now." It's not a meaningful concept in general relativity.