The way POP can work by default means that once a message is downloaded it will no longer appear in the original mailbox (I.e. Gmail checks for new messages, grabs them from pop.Tom.com and now the inbox in pop.Tom.com is empty).
If you found an old client in one place, I suspect you have another one that is grabbing your mail.
To confirm, change your pop password, update it in your gmail settings and then send a msg to the pop mailbox. If it works, you'll need to track down the client that has been downloading your mail.
Gmail has a notion of "me" as a sender and recipient, which is why searching your mail "from:me" works.
If you choose "Treat as alias," Gmail will treat the other address as "me" in addition to your main Gmail address. If you untick "Treat as alias," then it won't. Before this feature was added, all "send mail as" addresses were treated as aliases, or in other words, treated as "me."
There are a few minor repercussions. For example, if you send a message to "me," Gmail will put the message into your inbox. So if you send a message to address B, then Gmail will put it in the inbox if B is treated as an alias, but will not put it in the inbox if B is not treated as an alias.
"Treat as alias" does not affect whether your other address shows in the headers; that feature is controlled by your choice to use an SMTP server for the other address. It will also not affect specific searches for the other address, or your default reply address (e.g. "Reply from the same address the message was sent to").
You should use "Treat as alias" if the other address represents your own personal identity. You should not use "Treat as alias" if the other address represents another person (such as your boss) or a mailing list.
When you choose to use the SMTP server for the other adress and do not check the 'Treat as alias' box, a bug in gmail occurs. All mail sent by you as account B will appear as sent 'to:me' in your Sent Mail box. See this thread. While this bug remains unfixed, it is highly recommended to use "Treat as alias".
Best Answer
It seems that they've disabled the feature. That setting used to lie here:
But no longer exists.
I've searched the setting in Gmail but no luck.
In Inbox, there's a "Help & Feedback" option on the left. You can request that they reinstitute the feature there.