When do I stop a game for weather, and how is a suspended game resumed?

Once the game begins, the umpires alone decide whether it stops for weather or field conditions (Rule 4.01(d)). A local league rule can't override that.

Lightning: Little League's lightning guidelines say to stop and get everyone to a safe place when lightning is seen, thunder is heard, or the flash-to-bang time is 30 seconds or less. Dugouts and shelters are not safe. Wait at least 30 minutes after the storm before resuming. The guidelines recommend a separate "lightning monitor," not an umpire or coach, to watch conditions.

Resuming: a suspended game picks up from the exact point it was halted. It's treated as the same game, so mound visits already charged carry over, and the pitcher of record may continue if still eligible. Before you leave, make sure the scorekeeper has recorded the score, outs, runners and count.