What is a "Double Meet"?

I believe you essentially have it right. A "double meet" could also mean one eastbound train waits on a siding while two westbound trains pass, one after the other.

I' not exactly sure what you mean by your second question, if you are talking about the railroad cutting out the signal system and running everything under "manual block" rules, that happens from time to time. In passenger territory it will happen when there is something wrong with the signal system and they'd rather not delay the trains further.

In American practice, where no junction or other point of conflict is involved, a signal equipped with a number plate normally changes upon the passage of a train to an indication called Stop-and-Proceed (usually red, either alone or above another red), which permits a following train, after stopping at the signal, to proceed at Restricted Speed, which is very tightly defined as prepared to stop within one-half of sighting distance of any obstruction (which means that if you hit something you were by definition exceeding Restricted Speed). This rule would cover a double-meet situation without the necessity of departing from the existing signal system. I believe I read that on DB a home signal equipped with a panel of White/Amber/White/Amber/White instead of White/Red/White permits a train to pass the signal at stop prepared to find the block occupied, but that was a long time ago; what is the situation today?