Regarding the umpire's role and pay:
The umpire's jurisdiction doesn't begin until the home plate conference, so the plate umpire can't call it before then.
As Noumpere said, the coaches should be talking to the umpires beforehand to avoid a situation in which the plate umpire accepts the lineup card and immediately calls the game.
Different associations have different contracts with different leagues and schools regarding umpire pay in these situations.
In the associations I've been in, umpires receive a partial game fee or a travel payment if the game is cancelled within two hours of the scheduled start or after the umpires arrive. Once the game starts, however, umpires are due a full game fee regardless of when the game gets called.
A game fee is a game fee is a game fee. We don't get extra money for extra innings or waiting through rain or lightning delays. And we don't get less money for mercy rules or postponements.
Situations like this usually occur in rec or travel ball. High school coaches are protective of their fields, and ADs are protective of their budgets, so cancellation decisions for scholastic games usually get made early.
When I find a sloppy field at a game site, I make a pre-game round of the field. If it's hopeless, I tell the coaches they are already on the hook for half our game fee but they can save the other half by admitting defeat before I accept the lineup cards.
Usually they acquiesce. Sometimes, they push back a little.
"We really want to get this game in."
"And I really wanted to officiate this game, sir, but we're not going to play this game on this field. The weather has already won."
Nobody has ever let it go so far as to insist on starting just so I or my partner can stop it.
Probably the only argument an umpire can ever win with a coach.