how often will public function __construct() run?

PHP is shared nothing architecture.
There is no "run-once" type activities.

> What type of testing does your book address?

My book explains how to write PHPUnit test code, and
Codeception acceptance testing for CodeIgniter 3.x.

> I am having trouble with Composer. Does that preclude using your testing procedure?

I don't know. But the book is not about Composer.

And the book is a bit out of dated. You can learn testing
concepts and basic knowledge, but you will need to refer
to the manual of each tool to build the latest development
environment.

Software in the book:
• PHP 5.5 (You can use PHP 5.4 or 5.6)
• CodeIgniter 3.0
• ci-phpunit-test 0.10
• PHPUnit 4.8
• Codeception 2.1
• Selenium Standalone Server 2.48

ci-phpunit-test | CodeIgniter Testing Guide | CodeIgniter 3 to 4 Upgrade Helper