Can I change the name of index.php?

Codex is your frined – Create a Static Front Page

Sure, if you don’t want to use the WordPress blog stuff you installed.

Yes, it was a sarcastic answer.

In my defense, this last update was the first time you used the phrase subdomain.

Of course you can put the plug in a subdomain — or sub-directory — without having impact on your main site/domain.

Just install WordPress anywhere but in your main site. As you mentioned, put it in a subdomain; or as I mentioned, you could also put it in a subdirectory of the main site.

Check out Moving_WordPress for some help there.