You definitely have enough prerequisites to read Weibel, especially since you've already seen some basic homological algebra. The toughest part in terms of category theory for you might actually be getting used to abelian categories, but in most places you don't actually need to work directly with abelian categories, and you can use the embedding theorem (so essentially you prove things as if you were in the category of modules). So category theory probably won't be a block for you.
The good thing about Rotman is that he has more material on specific examples, and he goes into more detail.
Ultimately I recommend reading Weibel's book (however, as a disclaimer I have only read Weibel and just glanced at Rotman). Reading Rotman you will have gone through nearly seven hundred pages (Weibel: 415) and still not have seen Lie algebra homology, Hochschild and cyclic homology, and perhaps most importantly for someone who might be interested in topology, simplicial methods (the highlight of which is the Dold-Kan correspondence). To me this seems unacceptable. Rotman also places the chapter on spectral sequences at the end of the book and thus you don't get much of a chance to use them throughout the other chapters, such as group cohomology.
Finally, although Weibel's book has exercises, you might want to supplant them with a few more from either Rotman or Hilton and Stammbach.
Motivation for the Jacobson radical
Well, it sort of proves its own usefulness by being at the heart of so many algebraic theorems. But if you insist, there are a few good reasons that it is interesting.
For one thing, it is the largest ideal such that R/J has “the same simple right(/left) modules.” Looked at another way, it is the set of elements that don’t tell us anything about the simple modules (since they annihilate all simple modules.)
Motivation of the Jacobson density theorem
The Artin-Wedderburn theorem classifies which rings are like vector spaces in that they split into direct sums of simple modules. In particular, it says that simple Artinian rings are rings of linear transformations of finite dimensional vector spaces over division rings. What could be more natural? This type of ring is studied by undergraduates in linear algebra.
The Jacobson density theorem extends this result on simple Artinian rings to a larger class of rings called right(/left) primitive rings. A right primitive ring is a ring with a simple module with trivial annihilator. Rings of linear transformations of (right)-vector spaces over division rings (any dimension) are all right primitive, but this actually is a proper subset of all such rings. All of the rings that are “dense” in such rings are also right primitive.
The adjective “dense” is fitting because it is used in the topological sense. $R$ being dense means that (under a particular topology on the ring of linear transformations of a vector space over a division ring) that every nonempty open set contains an element of your ring $R$. In particular, given any linear transformation, there is a sequence of elements of your ring approximating that transformation.
Why do modules play an important role?
For you, approaching from the angle of representation theory, the most convincing thing might be that the F-representations of a group G are in one-to-one correspondence with the right $F[G]$ modules.
Really, there are lots of other reasons that modules are informative about their rings. That is basically the premise of homological algebra. You might check out Anderson and Fuller’s Rings and categories of modules
I feel the proof of some theorems are tricky and elaborate
Where is the rule that the theorems are simple and straightforward? What do you imagine they should be? Of course, this illusion of complexity in any field usually dissipates with growing exposure to the field.
If you really want to understand a particular theorem better, just find it in as many books as possible and compare the proofs and expositions. Usually this makes things come together faster. You should really check out Jacobson’s Structure of rings and Basic abstract algebra I+II since he was a pretty good expositor on structural theorems.
I can’t see the big picture of ring theory
Well, very few people can say that they understand the entirety of their field of mathematics. Most fields are just too large now. The classical structure theorems that you have already mentioned are nice examples (but hardly the pinnacle) of structural ring theory. If you want some nice books on general ring theory check out Lam’s First course in noncommutative rings or Faith’s Rings and things.
Should I read Hungerford again? Or just continue studying representation theory?
Whether or not you’re satisfied with the ring theory before you move on depends on your temperament and needs. Personally speaking I haven’t bothered referring to Hungerford since my qual exams. I never found Hersteins’ book useful, and I haven’t had the pleasure of coming in contact with McCoy’s book yet. On top of the books I’ve already mentioned, you might like Carl Faith’s or Louis Rowen’s volumes on ring theory as general references. If you check out the content on the Artin-Wedderburn theorem and the theory of right primitive rings in all the books I mentioned, I think you’d feel much better about the theorems, but spend your time as you see fit, of course.
Best Answer
There are many on this site better qualified to answer this than I, but I'll take a shot at it. Representation theory is a quite broad subject and depending on what exactly your interests are, you may need more or less background.
An afterthought: representation theory is such an enormous subject that it is simply impossible to say what the prerequisite topics are for "all of representation theory." There is much more to be said than what I have written, and there are a lot of major topics I know little about. However, the following things will get you started comfortably. Of course, you should not spend too much time "learning prerequisites" as there will always be another prerequisite. You should start learning something you are interested in, and learn the other things as you need them. Of course, a solid grasp of linear algebra is absolutely essential, so that is perhaps one fundamental you should not neglect.
• Finite Groups:
Linear Algebra: Jordan Decomposition, Inner Product Spaces, Dualization
Multilinear Algebra: Tensor product, Exterior Product, Symmetric Product, and the various identities relating them are quite important. This is not so much a field unto itself, but you will need to familiarize yourself at some point with identities like $V^*\otimes W\cong \operatorname{Hom}(V,W).$
Finite Group Theory: Lagrange's Theorem and other basic properties of finite groups, Conjugacy Classes, Orbit-Stabilizer theorem, examples of finite groups like $S_n,D_n, A_n$ etc. The Sylow theorems might also be useful.
Basic Ring Theory: There are some simple ring theoretic constructions that naturally pop up, like group rings $\Bbb{C}[G]$ and so on. Just having some familiarity with rings should suffice.
• Compact Lie Groups: (Lie algebra theory is needed in some parts here, and is arguably part of the theory.)
Manifolds: you should know what an abstract manifold is, smoothness, examples of manifolds, differentials of maps, and subsequently examples of Lie groups like $GL(n,k),SL(n,k),SO(n), SU(n),O(n),U(n)$, etc. where $k=\Bbb{R,C}$.
Linear Algebra: as above, also you should know about bilinear forms and what it means to preserve a bilinear form e.g. that $O(n)$ preserves an inner product
"Modern" Analysis: The Peter-Weyl Theorem requires some analysis background, so you want to learn about Haar measures, function spaces, etc. It might also help here to know a little bit about the Fourier transform I believe.
Riemannian Geometry: This can help you understand things like the exponential map on Lie groups in a different way, and there are some elegant proofs of results using notions of curvature. (Probably more optional than the others.)
Algebraic Geometry: If you want to learn results like the Borel-Weil-Bott Theorem, then knowing about sheaf cohomology on $\Bbb{P}^n$ (over $\Bbb{C})$ is essential. (This is also optional in some sense.)
• Finite Dimensional Lie Algebras: I'll suppose here you work over an algebraically closed field of characteristic $0$, otherwise there are more things to say.
Linear Algebra: Jordan Decomposition, Inner Product Spaces, Bilinear Forms, examples of matrix Lie algebras like $\mathfrak{sl}(n,k)$, etc.
Algebraic Knowledge: Actually Lie algebras have fairly light pre-requisites, but the beginning of the theory is mostly linear algebra in conjunction with structure-type theorems reminiscent of those from Groups and Rings. So, having some knowledge there would not hurt.
Basic Lie Groups: Knowing how Lie Algebras arise as tangent spaces of Lie Groups might make things feel better motivated, and also clarify some of the constructions.