I have read through the first few chapters of Spivak, however my personal preference is for Apostol's Calculus. It's also a very rigorous approach, and a very well respected book, however it starts more gently than Spivak's. With Spivak's book, the problems start out extremely hard, and get easier as the book goes on (mostly by getting used to his style, not objectively). With Apostol I was able to understand and answer all the questions in the first few chapters much more easily, and then I saw the difficulty increase a bit; however it increases progressively throughout the book. Many of the problems in the introduction of Apostol are exactly the same as those in Spivak, however the order and context that they are presented in leads you to the correct method for proving them, whereas Spivak's are more isolated.
There are many great discussions about calculus books on other forums, such as The Should I Become a Mathematician? Thread on Physics Forums.
I agree wholeheartedly with mathwonk's statement that, although the books are difficult, reading different approaches and going over them multiple times is really what gives you a deeper understanding of calculus. Mathwonk also mentions that most students find Apostol very dry and scholarly, where Spivak is more fun; however, I have not found this to be the case. I have worked through every problem in Apostol's Calculus through chapter 10 so far, and it has been a joy (most times). As an added bonus, Apostol's Calculus covers linear algebra as well, and the second volume covers multivariable calculus. Spivak's analogous book, "Calculus on Manifolds", is known as an extremely difficult text, and is commonly used as an introduction to differential geometry (indeed, his comprehensive volumes on differential geometry mention Calculus on Manifolds as a prerequisite).
The choice of book should also reflect your future interests. I am a computer programmer currently, and am looking to go into mathematics exclusively. It sounds like you are still melding the two. I would say that Apostol's book might serve you a little better in this respect as well, as it is slightly tilted towards analysis, whereas Spivak's is tilted towards differential geometry. For instance, Apostol introduces "little-o" notation, a cousin of "big-O" notation which is used extensively in computer science. That being said, Spivak has been described by some as a deep real-analysis text more than a calculus book, so you would still deeply cover all the fundamentals.
Another set of calculus books which I own and are held in high regard are Courant's. My brief skim of them, as well as other's comments, suggest that they are more focused on applications perhaps than some of the other books. Apostol's is still, in my opinion, very well peppered throughout with applications; many chapters contain a specific "applications of ..." section which links the theoretical concepts you just learned with the applied use of those concepts.
My only exposure to Courant's expository style comes from his excellent book What is Mathematics. This is a book I would strongly recommend reading regardless of what calculus book you choose. I cannot praise Courant's lucid writing highly enough, and look forward to working though his Calculus texts in the future.
I think that you would find Apostol's book sufficiently rigorous, as well as extremely intuitive. I also am a musician, and coupled with my computer programming experience it seems that perhaps we think alike. Whatever book you choose, recognize before you start it that you are running a marathon, not a sprint.
I think we can reasonably state that an ellipse is both $1)$ existent and $2)$ not a straight line.
The reality is that, if we are $100\%$ rigorous in everything we say, any textbook would be thousands of pages long. Each statement would have to be proven from axioms, and nobody would want that. There are certain reasonable things we can and must assume.
I also will say that I understand the frustration. Sometimes textbooks are frustratingly casual where you don't want them to be. But part of writing math well involves knowing which things are okay to exclude.
Best Answer
Here's a proof that an increasing function defined on $[a,b]$ is integrable. https://www.math.utah.edu/~yael/3210_public/exams/Integral.pdf . Assuming increasing means strictly increasing, it is also one to one so $f^{-1}$ is also defined, and is easy to be seen to be monotone as well, thus is integrable as well.. A monotone function on a closed interval is also trivially bounded, by $f(a)$ and $f(b)$