If I set up Apple Mail, will it delete my emails? I was trying to set up Apple Mail on my Mac because someone emailed me a .eml file that I seem to need Apple Mail in order to open. When I tried setting up my Gmail account in Apple Mail, it asked me to give permission for the app to read, edit, and delete emails.
What I Want To Know: Under what circumstances would Apple Mail actually delete my emails from a Gmail account that I set up with it? Would it only be when I manually delete them in the Apple Mail app?
The .eml file is a mail file that is based on the MIME RFC 822 specification.  It's basically a text format that pretty much all Mail applications can open

Outlook (macOS, Windows, iOS, Android) / Outlook Express (Windows Only)
Microsoft Mail (Windows 10)
Apple Mail (macOS)
Mozilla Thunderbird

Regarding your question, you don't need to setup a Gmail acccount (or any account for that matter) to read a single .eml file.  If it asks you to set one up, just skip it.
You can also open a .eml in your browser.  Simply rename the file with an .mht extension.  Assuming the file is on your Desktop, using Terminal
% mv ~/Desktop/foobar.eml ~/Desktop/foobar.mht

Then open with your browser.  If it's only text, any browser will work, but if it has embedded images, Chrome or Microsoft Edge will natively display it (I prefer Edge over Chrome as it uses the same engine, but I distrust Microsoft less than I do Google).
Microsoft Word will natively open both a .eml and the .mht.  If you don't have Word, you can open it with the free web version of Word.  You just need to create a free Microsoft account.
As an example, I got an old .eml file (an Uber receipt) and tested.  It opens natively in

Outlook for macOS
Apple Mail
Microsoft Word
Edge/Chrome (as a text file)