I'll try to answer your questions based on what I have observed about those subjects.
1) Topology as a separate subject, I wouldn't. To me, topology seems to be a spin-off of the geometrical investigations of the 1800's. There are alternative geometries, hyperbolic and all that but also inversive. Inversive geometry treats lines and circles in the same way because an inversive transformation transforms lines into circles (and vice versa). So there is a kind of forgetting that lines and circles are distinct; if we pretend they are the same, what insights can be found or what savings can be made?
And topology seems to be a similar thing, we forget that a cup is not like a donut, they both have a hole and one continuous body, let's pretend they are the same. Now what insights can be had, etc. So essentially one is taking about really esoteric things that are far removed from any immediate purpose.
And for example, what worries me is reading that knots are related to topology. Knots were obviously used throughout history by sailors, and renowned sailors like William Parry, probably the most respected English sailor in history, set out on incredibly dangerous journeys for years at a time depending on their knowledge and use of knots. And obviously that was long before there were any topologists. So what could topology possibly add of value to that subject?
Of course I realize that one never knows when something in pure math will become useful. Cryptography is the most well-known example, that number theory has become very useful indeed, even critically important for banking and all that. And clearly there has been a use of manifolds for general relativity and cosmology. But I'm of the opinion that the pure mathematicians should deal with the pure math while the rest of us, until that math becomes applicable, should leave it up to them.
For these reasons, I wouldn't learn topology as a separate subject. There is also the reason that I often see given on this site, that topology is complete, half a century old and stagnant. So surely it is better learn it as you need it in other subjects.
2) This
https://www.amazon.com/dp/0387901108/?tag=pfamazon01-20 book looks interesting to me. I always like advanced books that are clear, they usually work well for a survey because they will say things like "We take for granted that ... (*), now ..." where the starred content forms a very nice survey.
But I think that is too advanced because you have chosen two undergrad books. Axler I know, it is very dry indeed. If you try to prove all the theorems and do whatever research you need to achieve that, you will get a lot out of it. And many people do like it. But it's black text on a white page unrelentingly, with little variation.
The Halmos book ("Finite-Dimensional Vector Spaces") I don't know but it looks well written, and I know that Halmos will be very clear with his definitions, so probably I would choose that one without much hesitation, it will be good and has many glowing recommendations.
3) Would I choose something other than Rudin? Yes, absolutely and without reservation. Rudin is a wasted exercise. I think he was trying to write a Bourbaki-style hyperelegant book, something for really bright students, but I think he failed in his aim. What he ended up creating was math written on the surface of Swiss cheese, where the holes obscure the meaning.
The limited experience I had looking at his Principles book was the following: looking at a proof, deciding to study it in detail, realizing that there were definitions unknown, looking back to find the definitions, seeing that they relied on other definitions, looking back to find those other definitions and finding that they were taken for granted. It's just too patchy to be anywhere close to usable for self-study.
What book though, this one looks nice to me:
https://www.amazon.com/dp/0387974377/?tag=pfamazon01-20. It includes metric spaces but not immediately, as well as some n-variable content and Fourier series. If that is too slow for you, I don't know what to suggest. If I think of something, I may reply at some later point.
