I would really recommend that you plunge in and seriously read Aluffi's Algebra - chapter 0. The book will cover many many topics in abstract algebra including group theory, ring theory, field theory, as well as more advanced material like homological algebra.
The book is marvelously written which is a reason on its own for reading it. But, since you are looking for the category theory perspective this book is really what you are looking for. It does not assume any category theory, but instead develops parts of at as you go along, exemplifying everything with the algebra being developed at the same time.
Depending on you level of comfort with abstract ideas, you might find that you want to reinforce reading the book with reading a more elementary text on group theory. Rotman's Group Theory is excellent.
I really don't like Fraleigh's book, though I know it's popular. In my opinion the order in which things are presented makes little categorical sense.
The nice thing about Aluffi's book is that when you finish it you can truly say that you know the chapter 0 of modern algebra. It really gives you a very sound foundation of all of modern algebra.
I will respond directly to this part of your question.
I would be happy if I could avoid such topics [analysis], but I don't know what type of mathematics is studied in a graduate degree level, so this leads me to the following questions:
What topics of Abstract Algebra should I study in depth ? what topics in Abstract Algebra should I be familiar with their basics ?
Are there any topics in analysis, topology etc' that are likely to be needed for answering a graduate degree level type of questions ?
What should be the focus of my work, should I try to do many exercises within the text, or focus on the proofs and the theory ?
Are there topics in Abstract Algebra, or other in other areas that I would need to know (maybe topology ?) that I can skip some parts of (mainly non-core topics that are hard to learn) since they would probably not help me (and due to lack of time) ?
I have the book Abstract Algebra by Dummit and Foote to study with, as well as books in other area of mathematics such as Topology by Monkers that might help me with this goal.
Firstly, I want to mention that unless you are absolutely certain that you are going to specialize in pure group or ring theory, then you will need some analysis. In fact, you'll probably need a lot of analysis. Explaining why is a bit more complicated. The short version is that almost every area of math relies or is at least informed by analysis, algebra, and topology; this is why most graduate programs (in the US anyway) require these as either graduate classes or graduate entrance exams or graduate qualification tests, etc.
To expand in a slightly longer way - calculus is pretty interesting, and lets you do a lot of things. A common thing that mathematicians do is put measures on weirder spaces so that you can have some variant of integration. In number theory (even algebraic number theory, which is often the same thing as algebraic geometry, which is often the same thing as commutative algebra, which is just algebra and group theory), we really like having measures called Haar measures on matrix groups, like the $GL(n), SL(n), Symp(n)$, etc. This lets us do integration on these groups. So we study functions invariant under actions of these groups, or functions invariant on certain cosets of these groups that behave nicely under ring translation, or some similar idea. And one way we do this is to integrate them, or consider an integral over a weighted average of a function across the cosets our function is invariant over (read: Eisenstein Series for example), to extract largely algebraic information about number fields. Or we consider representations (as in representation theory, which I clump into the larger algebra domain sometimes) and analytic extensions of representations. Everything I've mentioned here requires a certain comfort with topology, analysis, and algebra.
This is to say that algebra mixes quite a bit with analysis in many ways. You would really benefit from having a good understanding of analysis and topology. In particular, don't focus solely on algebra. The other answer says this a little, but I am going to emphasize this a lot. It is very important to understand analysis and topology, unless you are going do limit yourself to pure, remote group theory. And even then, I wouldn't recommend it.
But back to your question at hand about algebra:
I would prescribe a path into algebra. In a comment on the other answer, you mention that you know groups, ring, fields, Galois theory. Cool! You also say you have Dummit and Foote (by far my preferred introduction to group and ring theory). Then I suggest two paths:
Go learn more about whatever parts you liked most. Sylow theorems interest you? Try to learn your way through Burnside's theorem. You like Galois theory? Pick up some infinite-dimensional Galois theory and try your hand. Maybe you already know that? Go pick up some algebraic number theory text - as an intro algebraic number theory text builds nicely on basic field theory and Galois theory, and suggests further paths. To be fair, I'm biased - I'm a number theorist. The important thing is that you go and dig deeper into things that interest you.
Pick up Atiyah and MacDonald's Commutative Algebra (hopefully from a library, as they're proud), and do your best at all the exercises. This is the 'natural' extension of what to do next, and it's the real path into a serious interest into algebra in my opinion. I say that you should do all the exercises because this book is famous for having really important lemmas and theorems in the exercises as opposed to the exposition. This will also really set your group theory and ring theory in stone, and you have Dummit and Foote to fall back on if you need. If you know this already, you should next go to Lang's Algebra (quite a bit, scary thing - take a look at it first), Matsumura's Commutative Ring Theory (much, much, much higher than Atiyah MacDonald, even though they have essentially the same name), or Eisenbud's Commutative Algebra (also harder than Atiyah MacDonald, but designed for people interested in algebraic geometry - if you don't know what that is, look it up).
I'd like to add one more thing about your (3) - the problem with learning the proofs and theory is that there is no reason for them to stick on their own. You might open up Atiyah MacDonald and understand everything you read, for example. But I wouldn't expect much of it to last, unless you use it. So a good general philosophy is to read and try to absorb, but then do exercises to let it solidify. Well written exercises require you to build on the text, both as a review and to build intuition.
A hard problem is knowing how many exercises to do. Too many, you waste your time. Too few, you'll forget much. But this is sort of moot, as it's hard to know what problems are useful or good to do before you actually do them, and in some texts some problems are much much better for you than others. For this, I advise you to ask your advisor (or find someone who can provide some sort of guidance) for direction once you have an idea what sort of things you want to learn about.
Best Answer
Paolo Aluffi's "Algebra : Chapter 0" is a kind of introduction to algebra in the categorical way. However, I would recommend Isaac's "Algebra A Graduate Course" despite it not dealing with category theory.