Why can't we expand the definition of the system until momentum is conserved? 
So, here a ball initially moving with a velocity v and collides inelastically with a pivoted rod. I've learned that this is a classic example for the conservation of angular momentum. My question is why can't the conservation of linear momentum be used here? I've asked quite a few people this, and the general answer is that the pivot supplies an external force and hence we can't use the conservation of linear momentum. But don't we have the liberty to define our system? Why don't we try including the pivot and the earth also in our system, that way all the forces are internal and hence we are allowed to apply the conservation of momentum?
 A: It depends on what question you are asking. If you want to know the angular velocity of the rod then conservation of angular momentum is much easier to use, as you have noted. Even if you are interested in what force the pin supplies, I would think that using angular momentum conservation would still be good to start with. 
You are right, we can make our system "larger", but it's not very useful. As the rod rotates about the pivot it's linear momentum is changing, so the momentum of the earth will be changing as well. So while linear momentum is conserved, the momentum of each part of the system can still be changing. It's not "wrong", but why do more work than necessary? I think there is also the issue of that it would be hard to say how the linear momentum initially gets transferred to the Earth-rod system, since the collision is not instantaneous, so the rod is already moving, changing direction, and being acted upon by the pin while the ball hits it. I would have to think through it more though, but you can already see the complexity of trying to consider linear momentum.
A: Momentum is always conserved. It's a universal law of physics. But applying it to a system that includes the earth is not helpful here, because you don't know the earth's final momentum, and you don't want to know the earth's final momentum. That makes conservation of momentum true but useless in this situation.
A: Yes, if you solve the problem exactly, the momentum of the full system would be conserved--but it is fairly complicated to solve the problem exactly and thus, we use an approximation.
The approximation is to consider the Earth to be infinitely massive--in particular, it can absorb/release any finite amount of momentum without having to attain a velocity. Thus, the ontology in our approximation involves the existence of a reservoir of momentum and thus, momentum is not conserved. Interestingly, by the virtue of the same approximation, even angular momentum of any system involving the Earth is not conserved because an infinitely massive Earth also acts as a reservoir of angular momentum. 
But, the flip-side of this apparent menace is that the Earth simply ceases to have any dynamics (which was the purpose of the whole approximation, to be honest) and we don't need to consider a definition of the system which assigns dynamic degrees of freedom to Earth--because it simply has none in this approximation. Now, we only treat the rest of the system (i.e., rod and the ball in your example) as our true dynamic system and treat the Earth as the environment which affects the dynamics but cannot be affected in reaction.
Now, for such a redefined system, which conservation law holds and which doesn't depend on how it interacts with the reservoir (the Earth), because, potentially, a reservoir can influence the system in a way so that no conservation law might hold for the system! Thus, we need to analyze a situation in its own right. Here, in your example, given the way in which the Earth interacts with the system is through a force that always passes through a fixed axis and thus, the conservation of angular momentum holds for our system about that axis. Notice that since the momentum is not conserved, even angular momentum is not conserved about any other axis!
