If you want to change the name of your Blogger / GMail / Google account, you will have to create a new Blogger / Google account, and transfer control of assets (such as Blogger blogs) to the new account. This is inconvenient - but it forces you to acknowledge that you are creating a new account.
If you setup a Blogger account that's based upon a non GMail email address, Google should send a notice to the non GMail email address, requesting ownership confirmation. That's how many web services verify email addresses used to register their services. Google, however, makes it easy to use a non Google email address, when setting up a Blogger / Google account. Just enter the non GMail email address, and you're done.
If you entered the address of a non GMail email account that you do not own - or that does not exist, this is not a problem, during the Blogger account setup process. Maybe later, this will be a problem, for you - or for somebody else. But it's not a problem, right now.
Now, you have a new Blogger account, and you can accept membership in someone else's blog, create your own blog, and / or Follow the blog(s) that interest you.
What if you rename a non GMail email account that you own, or change the password for that account? Any changes you may make to any non GMail email account will not be mirrored in the corresponding Blogger / Google account - since these are two separate accounts.
If the non GMail email account is deleted, for any reason, you can survive, since these are two separate accounts.
- You delete the account, intentionally.
- You change ISPs, and lose the old email account.
- You graduate, or leave the job, where the email account is based.
When you try to sign in to Blogger, using an account that cannot administer a specific Blogger blog, you'll have more challenges. With Blogger never telling you what account you should be signing in, you'll never know why you cannot administer your blog. If you try to recover ownership of your blog, using an account that you do not own, you'll get nowhere.
What you will have to do is understand the issues, such as outlined here. If you don't understand these issues, you'll have problems that Blogger can't help you with, and that you can't solve.