Gmail – I can’t send emails or respond to emails. I click “send” and nothing happens.

Have you had a look at the Insert this signature before quoted text in replies and remove the "--" line that precedes it option just below the signature edit box in your settings?

It’s probably that Gmail is adding your signature after the quoted text when you click reply.

Also, Gmail automatically trims content when you view emails:

The "trimmed content" is typically quoted text from a previous message (identical content) and so they hide it so what's initially visible is only the NEW content. Most people don't need to see the message they just sent that the person is replying to. They just want to see the actual reply (new content).

From Google:

Signatures are separated from the rest of your message by two dashes. To see a signature in Gmail, click the Show trimmed content button at the bottom of the message.

Yes, this is possible using the Canned Responses lab combined with filters.

First, create two separate canned responses for each of the scenarios. Here's a good how-to for setting that up: http://time.com/12436/quick-gmail-trick-pre-write-email-messages-with-canned-responses/

Then you need to setup two filters:

from:(domain.com)

And:

from:(-domain.com), -(category:social), -(category:promotions), -(category:updates), -(category:forums)

And check the option to "Send canned response" and select the corresponding response.