Yes, ships in the Age of Sail would have something like a stove or a hearth for cooking, either set directly on bricks on the deck or raised over the deck. The latter arrangement can still be seen on board HMS Victory; USS Constitution has a stove that sat directly on firebricks (n.b. the photo is contemporary; the large fan and fire extinguisher are not original equipment).

As to your question about fires on ship: fire was (and is) feared and quite deadly, and sailors took care to use only sealed lanterns for illumination belowdecks. The powder magazines were illuminated by lanterns with thick glass set into the walls of the magazines, with the candles being lit from outside. Ships catching fire during battle was always a possibility, and there were several instances during the Napoleonic wars when a ship caught fire and blew up during battle (not nearly as often as the Total War series would make you think). HMS Queen Charlotte caught fire in 1800 after someone apparently stacked hay on a match tub and it caught on fire; the crew was unsuccessful in stopping the flames and it blew up with about 680 casualties.