- Are there any security issues with creating an open source project that will run on a website that you host?
Not any that wouldn't be a concern for any other site. You should be extra careful to keep sensitive data (API keys, passwords, etc.) out of the source repository though. It's public, so more of a concern than closed source code.
- How do you host a website for an open source project? Who pays for the server fees?
The same as you'd host any other site. If you're running the site, you pay for hosting, domain, etc. If there's a really active community, you might get enough donations to cover it, but there are lots of low cost options. I self host and my site costs me about $80 a year, including server, domain, and tls cert. Having someone else host will cost you more, but you don't need to worry about much. There are also free options like Github Pages.
- Is it considered unethical in any way to display a webpage that was developed via Open Source on a domain that I own personally?
Please check the license, but the whole point of Open Source is to allow others to use your code. For what it's worth, I host a personal instance of Wekan. It's an open source web app and I have a copy of it running and available on the net.
As an aside, and answer to the title question, I made my website source available.
And a FLOSS project I used to be heavily involved in made our project site's source open.
So, is it common? I don't know, but I seem to have made it a habit.
One thing to note is that neither of these sites are very interactive. They're both mostly static content. There are no logins, no need for authentication/authorization, and the code does not make up the core IP of either project, so there's no big security benefit from keeping the source closed. There's nothing (to my knowledge!) that could be exploited by virtue of having access to the code. If either of these had security sensitive code, I would keep it closed.