Yesterday, MacWorld posted a useful article about Apple’s $29 iPhone battery replacement program.
All of your questions answered.
The article covers the usual FAQs. In particular, “How do I initiate a battery repair?”
As for the “How do I check my own iPhone’s battery life?” question, I downloaded and tried the coconutBattery macOS app and found the results interesting. Battery capacity was still okay.
Here’a a Washington Post video post “Replace your iPhone battery now. Here’s why” from January 26, 2018, which answers the “to replace or not” question.