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I would like a "good" book (not really introductory, not too advanced with good theory and exercises) on metric topology covering the following topics:
Metric spaces, open/closed sets, sequences, compactness, completeness, continuous functions and homeomorphisms, connectedness, product spaces, Baire category theorem, completeness of C[0, 1] and Lp spaces, Arzela-Ascoli theorem.
It may not be a full book but parts of books or lecture notes are also most welcome.
Look at Metric Spaces by E. T. Copson. It is volume 57 of the Cambridge Tracts in Mathematics series printed by Cambridge University Press. It should be available in paperback or hardcover. Here is a link.
You could have a look at Intermediate Mathematical Analysis by R.D. Bhatt. It does not cover all the topics that you mentioned, but provides an excellent treatment (with lots of exercises) for topics upto and including connectedness in your list.
Chapter 4: Metric and metrizable spaces.... in "General Topology" , by R. Engelking. You need to browse the Introduction to familiarize yourself with his notation (e.g. $\{x_i\}$ is a sequence), and be aware that for him, compact space means compact Hausdorff.... Chapter 3 is Compact spaces. It includes a beautiful short proof of the Stone-Weierstrass Theorem ( see 3.2.18 thru 3.2.21).... There is a wealth of extra material in the problems and exercises of this book.... But for $C[0,1], L^p,$ and Arzela-Ascoli, etc., I would suggest a text on functional analysis.
