If my 3rd party email changes, should I change the Apple ID to iCloud email?

I would change it to another 3rd party email address. Based on what I have read, I suggest you leave your Apple ID email address as a 3rd party email address because you have more flexibility if you need to change the email address. For example, if the Apple ID got compromised, you can change it to another address. I also have read of many people who were locked out of their accounts when they forgot their password. Since the recovery email address is the @iCloud email address, if you forget your password, you also can't access your email to get emails from Apple to recover your account. You can set a rescue email address, but for some people, this hasn't worked.

Note the restrictions in the article.

Apple ID Name Change.        

 What to do after you change your Apple ID or password - Apple Support    I suggest doing this before you change your Apple ID or password.

You are welcome.