Problems with running a 2nd installation of WordPress on my domain?

To avoid .htaccess files of WP installations, affect other WP installations, use the following file system structure:

hosting_account_folder
    ┌─────┴─────┐ 
public_html subdomains
           ┌────┼────┐
          sub1 sub2 sub3

In this scenario, mydomain.com points to hosting_account_folder/public_html, sub1.mydomain.com points to hosting_account_folder/subdomains/sub1, sub2.mydomain.com points to hosting_account_folder/subdomains/sub2, and sub3.mydomain.com points to hosting_account_folder/subdomains/sub3.

As you can see, none of WP installations, is a subfolder of any other WP installation.

Note: having a WP installation, as a subfolder of another installation, is not good for SEO, it is a myth. Always use subdomains!