They typically require supervision to make them forage from the berry bush once they've discovered farming.

The crops should regrow after approximately 4 hours on normal speed. There have been rare reports of occasional "famine" but the crops always come back at some point. I have found that leaving the game running with the Palm on AC power helps "straighten out" a wacky food situation if one occurs. The farm is getting some attention for the v1.1 update. Hopefully your crops will come in soon and you won't have to worry about it!