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tl;dr

Is there a good theory book to go by that covers (in the same chronological order) the concepts in "Problems in Mathematical Analysis" by Demidovich?

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I have about 3/4 of a year before I go to college and I'd like to complete Demidovich's book by then. I'm an ardent studier, and if need be, can work on math for 4 hours a day (the rest of my day is saved for other subjects), 6 days a week. I'd prefer a "rigorous" book, but not to the extent of Courant's "Introduction to Calculus and Mathematical Analysis".

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    I don't know, but I can tell you this: get used (soon and quickly) to use more than one book. It's almost impossible to find one single specific book for everything. Do your researches. Said this and you can start with Tom Apostol - Calculus. – 2017-01-30

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It is "Analysis basics" by Grigorii Fichtenholz. Its ISBN is 5-9511-0010-0 on my hardcover in Russian.

The book "Differential and Integral calculus by: Piskonov" is subset of those of Grigorii Fichtenholz from proofs point of view.

Demidovich's Exercises contains proof exercises, which are analogous to those in "Analysis basics", but those have got no counterpart in "Differential and Integral calculus by: Piskonov".

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I think that the book "Differential and Integral calculus by: Piskonov" is that book you want.

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In Russian math community the book that's usually considered as one the best published material on analysis is Vladimir Zorich' "Mathemtical Analysis", hope you can find enough tasks there:

https://www.amazon.com/Mathematical-Analysis-Universitext-Vladimir-Zorich/dp/3540874518