I do not know the history of the glorious title of this theorem and when I first learned it I wondered pretty much about the same things. It is not obvious that this statement deserves such a serious title. However, you have to keep in mind that when Noether proved it, the general understanding of these things was at a completely different level.
I would say to
Q1 that those applications are not simple. Especially Proposition 3 is a very important and at first perhaps surprising statement. This is completely unique to cubics and this statement is essentially equivalent to the group law on cubic curves. If you start there, isn't that an unbelievably beautiful statement that if you take a smooth cubic curve then you can define a group law just by drawing lines? Try to prove associativity without Proposition 3!
Q2 Probably this theorem today would not get the adjective "fundamental" and it would probably be hard to publish it in a good journal, but you must know other examples of fundamental theorems that now seem almost trivial. You have to place these in the appropriate era. I am not a historian so I will not do that, but let me point out that even if the proof does not seem hard at all, the statement is probably one of the first in a long line of important local-to-global theorems. In algebraic geometry we often have to rely on local data and observations. For instance, if you work on proper (e.g., projective) schemes you will not have any non-constant global regular functions. Sometimes you won't even have non-trivial line bundles. Hence it is important to be able to make global predictions based on local observations. Noether says that in this particular question you can do that.
As an exercise, that perhaps helps to appreciate the statement more, try to formulate what it means for the $\mathscr O_X$-module generated by the two homogenous polynomials.
Remark: see also Roy's comment about the Residue theorem and RR.