According to special relativity, two events can be a cause and effect of each other if those two events can be found in any lightcone.
You're not being explicit enough here, it may sound like you're claiming that which is the cause and which is the effect is "up for grabs". So to be clear and sure we're on the same page: two events are causally related if one is within the other's light cone, but it is an absolute fact that the event on which the light cone is centered precedes the event within its light cone - all observers will agree on that.
This contrasts with Galilean relativity, where two events can be a cause and effect of each other regardless of spatial distance.
No, this is wrong. In Galilean relativity, or what is sometimes termed "Newtonian spacetime" we have a universal notion of time. It means that to each event we get to attach a universally, observer independent, real number $t_E$, the time in which event $E$ occurred. So that if $t_{E_2} > t_{E_1}$ for example, it is forbidden by construction that $E_2$ preceded $E_1$ in any frame. Only causal relation in the other direction $E_1 \rightarrow E_2$ is possible.
However, this begs the question: given the geometric representation of special relativity, it is not very clear if space and time are indeed one, which forms spacetime, or if space and time are separate and just stuck together using some cosmic glue. So which one is it?
And is it possible to perform a experiment to determine if spacetime is only one thing or space and time as distinct quantities but glued together using a cosmic glue?
You'll need to be more specific about what you mean by "cosmic glue" if you want this question to carry any meaning. In other words, you need to provide a way to distinguish between a "truly unified space and time" vs. a "separate space and time that are glued by a third entity". As long as you can't say anything about the nature of that third entity and how one might detect it, we might as well apply Occam's razor here and assume it doesn't exist. For like in many other cases, we don't want to add invisible paraphernalia that only complicates the model but adds nothing novel to its predictions.
Regarding the "Update" section you've added in order to address exactly that, note that what you've written is just way too vague and handwavy to provide any kind of idea on how to actually test this idea. Just giving the analogy of Hydrogen as composed of more fundamental entities is not enough. You would either need to:
Suggest an experiment that you think can demonstrate space and time are two separate entities that are somehow "glued".
Explain why you think existing experiments already suggest that same thing, and even better, why you think the existing, working models, should also be interpreted in a way that suggests space and time are indeed "glued" in that way.
