Does an extension exist, or can someone explain me how to achieve it, that a customer can freely set the price for an item within a certain set of boundaries, i.e. "Pay what it is worth to you" ?
You can take a look at this Price Alert extension for Magento 2. The plugin allows users to enter a custom price on product pages as well as enables the email notification once the product price is decreased to the entered one.
Hope it will be of help!
you need to customize on the prouct page.
on checkout and cart page you can do using below link
hope it will help you.