Gmail – Implications of removing a Google+ account

On one of my accounts, the name was updated immediately after sending an email message. On another account, I just checked back after several minutes, and it showed the right name. So it seems that this is either a delayed process, or you must invoke some functionality (e.g. send an email) for Gmail to pull these settings from your account.

As you've noted, the Google support page has information about uploading profile pictures. Specifically, the page notes that you can make the choice as to the people that can see your profile picture:

You can select whether you'd like all Gmail users to see your picture, or only those who you've allowed to chat with you, using the radio buttons next to your uploaded photo on your Settings page.

• Visible to everyone means anyone who you email, or who emails you, can see your picture. If you have a public Google profile without a photo and you choose this option, your picture will be used on your Google profile and will be visible to others there.
• Visible only to people I can chat with includes users who have been given permission to see when you're online and to chat with you.

Additionally:

If you have a Google Profile, changes made to your Gmail picture will be reflected in your Google Profile.

Looks like the Gmail picture and Google Profile picture are linked. However, from my reading of the support documentation, your profile picture will only be visible if the email recipient that's reading your email is using Gmail.

One thing that you could check is to see if you've associated the non-Gmail account that sent you an email with a Contact. I have contacts that with multiple email accounts (both Gmail and non-Gmail). The picture for the Contact appears to automatically show the Gmail profile picture.