For instance, conservation of momentum, does it take time to propagate between two or more objects?
If it does, then there would be some moment that the momentum is not conserved.
If it doesn't take any time at all, since the law itself is information, then doesn't it prove that information can travel faster than light?
Conservation laws don't "propagate". They are inevitable consequences of symmetries of the dynamics by Noether's theorem, and the dynamics propagate with whatever finite speed they do.
See https://en.wikipedia.org/wiki/Continuity_equation - I was just writing some text there that I think helps explain.
Continuity equations are a stronger, local form of conservation laws. For example, the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy is fixed. But this statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.
If conservation of energy only said that total energy is fixed, and energy actually could teleport, then your question would be a very important question. But actually, every conservation law in practice is the stronger type, a local conservation law. Energy can only move by a continuous flow, and same for momentum and anything else.
A flow of energy, just like a flow of any other physical quantity, moves at or below the speed of light.
Conservation laws do not propagate instantaneously (or really at all), it really means that some property of the system does not change over time.
In the case of momentum conservation, the information about the collision propagates (at most) at the speed of sound in the medium. This is why you see the cars continuing along their path during a collision:
If the information of the crash travelled instantaneously, then the rest of the car would not move forwards after the front of the car impacts the wall, which is not what we see here. So no faster than light information travelling here.
Let's look at an example, electromagnetism.
The electromagnetic field (the combination of electric and magnetic fields) has momentum.
Charges have momentum.
The charge feels a force right where it is, a force based on the fields right where it is. This changes the momentum of the particle.
The field loses an equal and opposite amount of momentum and also does so right in the same place. (The field has momentum distributed in little bits all over space, so has momentum right there to give to the charge.) Technically there is just a flow of momentum from the fields to the charge because the charge is also just changing its momentum at a certain rate, nor all at once. So you can also think of it as the charges momentum being given to the field in a way where it starts to spread out through the field into larger and larger regions. Since momentum has a direction there is no objectivity about whether you lose $p_x$ or gain $p_{-x}.$
So momentum propagates through space, and can be stored in the electromagnetic field and flow through space via the flow through the electromagnetic fields (and yes, technically the fields have a momentum and a flow of momentum) in a conserved way up until it meets a charge at which point the momentum in the field is no longer conserved but the total momentum (field and charge) is conserved.
Momentum is conserved locally. And it can take time for momentum to get from one object (charge) to another object (charge). But when it does so, momentum it is still conserved in the time in between because in between, the fields have the momentum.
Same with energy.
Others have explained that they don't propagate. That's a real puzzler though, and the real question is how does the stuff (energy, charge, whatever) always add up when a transfer takes finite time?
The answer to figuring that out was a profound pillar of modern physics: the field contains momentum (etc.). So while an electron is handshaking with another charged particle to say "you go this way, I'll go that way", you have the further puzzle of relativity of time as there is no "simultaneous" in an absolute sense. One changes before the other, or vice versa, in different reference frames. The solution is that the electric field (electromagnetic, if you are using relativistic effects) contains the momentum that is not at the moment (in your frame of reference) ascribed to either particle.
I'm sure I butchered that, but you get the idea: fields are real things that carry these properties, not just a way of drawing maps.
All conservative phenomena (not laws), do indeed, take a finite time to propagate, therefore "information" can't travel faster than light. Also, just because phenomena take time to propagate, does not mean that the phenomena are not being conserved at any given instant of time. In fact, there is no moment, when the phenomena are not conserved!
