If I request a paper through my university library, must they pay a substantial amount of money?
You should not hesitate to ask the library for articles like this. The library most likely will pay a fee, but it will probably be less than the $40 for you to access it directly, and is an expected budget item for academic libraries.
There are costs associated with interlibrary loan (ILL) transactions, but they are minimal for articles that can be handled entirely electronically (for things that need to be physically transmitted, like books, and/or processed in some way, like print articles that must be scanned, the costs can be substantially higher). Since you found this article online, it probably falls in that cheapest category.
Many libraries also have agreements with publishers to "buy on demand" some articles that are requested through ILL. This usually happens entirely behind the scenes, so you probably wouldn't know the difference between such an article and traditional cross-library lending. It may cost the library more than ILL, but the library then "owns" the article so any future requests for that article will be at no additional cost, unlike ILL, which costs the same every time.
As an academic librarian, I can tell you that the library almost certainly has a budget for this kind of thing, and if it were my institution, I would want you to use this basic library service to get the materials you need to do your job (almost as much as I want students to get the materials they need). Nothing in a library is actually free*; making information accessible is the reason libraries—and library budgets—exist.
*Of course there are open access journals, donated books, etc., but someone had to pay for all of those, too, one way or another.
As to publishers: if the article you need was published by the same publisher as the journal you're reviewing for, there is a chance. Indeed, a few publishers grant temporary online subscriptions to reviewers. But otherwise, probably not.
Also, if you contact the editor and let them know that you need this other article to be able to review properly, and if the editor happens to have institutional access to the journal in question, and if they are not too busy, they might send you a copy of the article as a professional courtesy.
Likely not. Most university libraries have partnerships with other libraries. They will likely contact a library that does have access and request a copy. This usually comes with some limitations (for example they might only give you a paper copy, not a .pdf version) but they shouldn't pay anything extra.
I do not believe publishers will ask provide papers at request however. They likely expect you to do the above process if you don't have access.