If we assume that there's a universe that is devoid of causality, and if we assume that it isn't eternally static, would that mean that everything can happen because there are no reasons inhibiting anything from happening?
If the world were without causality then it need not change in any way. It might fortuitously behave exactly as it does now. This is certainly a logical possibility.
If the world were without causality but none the less followed probabilistic laws - exhibited probabilistic regularities - then we could easily get by counting on such regularities if their probabilities were high enough to allow a certain level of predictability and computability in our experience.
If the world were without causality and also chaotic, so that there were complete unpredictability and uncomputability of subsequent given initial states, then not only would our experience be unpredictable and uncomputable but 'we' would disappear as persons or mental continuants (in any sense recognised now) because our mental life would be subject to the same unpredictability and uncomputability. And whatever the proper analysis of matter and mind, of physical and mental events, our bodies would cease to be continuants since (again in any sense recognised now) there would be the same complete unpredictability and uncomputability of subsequent given initial states.
Life would be rather different, if indeed it continued in any form.
Philosophers and scientists who reject the concept of cause as incoherent or who simply deny the existence of causal relations can, of course, treat the question as on a level with 'If we imagine a world without phlogiston, what would it be like ?'
Robert W. Batterman, 'Defining Chaos', Philosophy of Science, Vol. 60, No. 1 (Mar., 1993), pp. 43-66.
The question needs a better definition of terms. What exactly is it that you mean?
If you mean a complete disconnect between events so that there is no chain of causality, then the answer is simple: The world would not "be" in any sense that we could apply. Consciousness requires causality, otherwise no thoughts would arrive from any inputs to the system. No connected thoughts would arise and no "I" would be able to form.
Most things in the world, especially life and anything connected to it, are strongly linked and kept together by causality. This is the underlying structure of anything we call a system.
If you mean a weaker version in which causality (A -> B) continues to exist, but preconditions are weakening (i.e. B can also happen spontaneously) then we need to discuss the frequency at which spontaneous events happen. As long as they are strongly dominated by deterministic events, you would simply have a lot of chaos, the exact nature of which depends on the scale at which spontaneous events happen (quantum scale? micro scale? macro scale?).
Causality is a concept, not something existing in the real, physical world. The closest you get is to look at physical processes which "cause" things to happen
Example: the constant pull of gravity on the apple weakens the bond to the tree, causing the apple to finally fall to the floor. It makes little sense to imagine an universe simply without the cause per se. You'd have to imagine an universe without gravity.
You then have to get rid of all physical processes that "cause" anything, ending up with an inert soup of stuff at best, and nothing at worst. In the previous example, instead of getting rid of gravity, you could get rid of the biochemical process which makes the apple prone to falling down, and so on and so forth. You'd still have to get rid of gravity as it causes an awful lot of things!
So, no, a world would not function without causality as expressed in physical processes.
It seems to me the absurdity to most things isn't with our experience of some state of affairs, but how we're conceiving of them. It doesn't seem absurd to say "The world would be different if the world were different"; we just accept that it would be and can imagine that such differentness is possible. It's when we attempt to say "If I think of the world differently, I would think of the world differently" that a fundamental problem occurs. That is, we can't seem to easily part with the notion that how we think of the world is actually tied to how the world is. Humorously, we think that "If I think of the world differently, then the world would change."
As it is, if one holds that there is a distinction between causality and orderliness, then we have the possibility of a non-causal reality yet explainable in reasonable, useful terms.
Without such a distinction, the question would be literally absurd and attempts to directly answer would be incoherent; but given reality, we at least know certainly that there is order.
Assuming you mean the philosophical misnomer "causality" meaning "being able to distinguish the past from the future" (as opposed to the physics meaning of the term), the question is ill-defined. The universe exhibits causality because it follows certain, very basic laws of probability -- high-entropy states of the universe are more probable than low-entropy ones, so when we keep observing more states, we're likely to observe higher and higher entropy states, which gives us a metric to distinguish the past from the future.
How exactly would you change these laws to produce a universe that doesn't follow the second law of thermodynamics? Until you precisely, mathematically answer this question, it's impossible to tell how your universe would look like.
If you mean causality in the physics sense, you would have a universe that behaves much like our own except faster-than-light travel is permitted.