Absolutely, you can delete the original admin user account now that you've set yourself as an admin. The process is fairly straightforward:
- Open System Preferences.
- Click Users & Groups.
- Click the lock and enter your password to make changes.
- Select the old admin account in the list of users.
- Click the minus sign below the list of users to remove the account.
The 'allow user to admin this computer' box being greyed out isn't a problem, as you've already established your own admin account.
If you're still having trouble or want to look at this in more detail, I'd recommend checking out this
link which provides a step-by-step guide on how to do this.