Again, no argument from me on that

Thanks for your detailed update, useful info.

No, an Outlook.com account—the new Hotmail. But it hardly matters based on your update.

Total long shot out of left field if you want to try it: Change your DNS provider to Google's—Primary 8.8.8.8, Secondary 8.8.4.4

Have a look in the Trust Center, see if you can add Gmail to something there—been too long since Win7/Outlook 2010 to point to anything specific. I've solved an occasional connection glitch by rooting around in the Trust Center.

Were they using Win7 & Outlook 2010?

Any chance emails from Google to you re your accounts could have gone in your spam and you missed them? I know, hugely unlikely with ~6 accounts—unless one of them was set as the 'Recovery Address' for all the others. Just yesterday my better two-thirds was locked out of one of her regular Gmail accounts while trying to access it from a different laptop. As her Recovery Address, I got the Google email with instructions to sort it out.

Another long shot: Could you have reached some kind of storage or transmission limit in Gmail? Unable to Send/Receive is symptomatic of full mailboxes and similar resource constraints.

Sadly, I'm flailing about on this one. There's evidence it can't be only Gmail, or your network, or Outlook—which leaves some weird interaction between 2 of those. Please post back if you discover the cause.