The highest degree of rigor would be achieved this way: Write down a list of axioms and a list of rules of inference. Start from an axiom and modify its logical formula using only one rule at a step and at each step clearly stating what rule you have used.
I'm not a logician and my description is probably not very good, but my point is that, whether or not this or some similar approach could be called "absolutely rigorous", it is highly impractical. At some stage in your development you know the law of associativity and need not be reminded of it every time it is used. At a later stage in your development the same is true about, say, the binomial theorem.
So, in practice, rigor is a relative concept. Proofs are written for the reader to check them. The goal of the author is to make this checking as quick and effortless as possible (this is often not true for textbook authors). To this end he must find the right amount of detail. The reader should not lose time by having to check four steps for a statement he could have easily understood in one step. On the other hand, the reader should not be forced to brood a long time over a statement that could also be written up for example with three intermediary steps each taking only a tenth of this time to check.
But this checking process depends on the reader. In my opinion, you cannot talk about rigor without talking about the "mathematical maturity" your average reader has. In a research paper proofs are considered rigorous that would be called handwaving or incomprehensible in an undergraduate textbook.
Of course, giving the right amount of steps in proofs is only one aspect of rigor. Another is not speaking about concepts you haven't properly defined. But this is relative as well. In a research paper about mathematical physics you don't have to clearly state the axioms for the real numbers. In a calculus textbook you do.
In your particular example: If you can expect your readers to know for example that the coefficients of polynomial functions are unique, your proof (or at least that part of it) is rigorous. If not, it is not, and you have to give some explanation.
You ask about when should you be satisfied that a proof is a proof. My opinion: if and only if you are sure you could, if required, fill in all thinkable intermediary steps and trace each fact you use back to the very axioms. If you are already asking yourself "is this really a proof?", it is most certainly not for you (though it is for Spivak, and, if he succeeded, for his intended readership). You have to break it down to steps that are completely obvious to you. In your mathematical development more and more arguments come to fulfill this criterion. For example, when you first learn about induction, you need to clearly state the base case, the induction hypothesis and so on. Once you have seen your share of proofs by induction, you are satisfied and often grateful if the author just writes "by induction we get...".
If you've worked through the majority of Apostol, then I highly recommend Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms. (Get it straight from the publisher; much cheaper than elsewhere). It is written very well, with the reader in mind. You'll learn linear algebra and some other requisites to start with. From there, you learn a bit of very basic topology and you end up doing some differential geometry-esque stuff later. There is SO much in this book... you could study it for a couple of years (especially if you dig into the appendix). Moreover, despite its rigor, there are many applications (that are actually very interesting).
As well, there is some courseware from Harvard that uses this book (look for Math 23a,b and Math 25a,b). And there is a (partial) solutions manual floating around.
You might also like to read:
-Set Theory and Metric Spaces by Kaplansky
-anything from the New Math Library (from the MAA)
-Particularly Basic Inequalities, Geometric Transformations, or Mathematics of Choice.
-the linear algebra in Apostol (be sure to get Vol 2 also!)
-the whole second volume of Apostol!
-Discrete Mathematics by Biggs (get the first edition!)
-Finite Dimensional Vector Spaces by Halmos,
-Principles of Mathematical Analysis by Rudin with this and this and these awesome lectures.
-Algebra by Artin is amazing, but hard! Enjoy these lectures. Vinberg's algebra text is supposed to be amazing and in a similar flavor to Artin (but a bit more gentle).
And you might also like these (great) reading lists:
-PROMYS
-Chicago Undergraduate Mathematics Bibliography
Best Answer
This isn't a complete answer by any means.
A couple of weeks ago there was a conference based on the work of William Thurston. There were several references made to an idea he used (I believe) of having levels of understanding. When you first meet something, you can read the theorems, and get a first level of understanding. But as you come back to it, in different contexts, seeing it from different points of view etc, you gain more and more insight.