Engine staling, thinking battery problem but lights still come on.

HI, on saturday I drove the car about ten minutes from my house to the grocery store went in for about 15 minutes then drove the car back took another ten minutes on my way home I could smell engine coolant and noticed smoke. When I pulled in the drive way I noticed alot of smoke comming from under the hood, after the car cooled my husband checked the car we noticed the side of the thermostat housing was completly missing. So on sunday we went and bought another thermostat housing unit and my husband replaced it as he had done in the past. He replaced the radiator cap and added antifreez coolant and radiator stop leak since we have had problems in the past, after all this was finnished he drove the car that night to and from his job located at the same grocery store about ten minutes from my house with no issue. But when he arrived home he told me he needed to add oil because the car was low on oil. He did so and showed me the oil cap and stated some water from the engine may have gotten into the oil from the undertightened oil cap, but it shouldnt be a problem and we would get the oil changed soon. So he went tot ake the kids to school, about 15 minutes after leaving going up a bridge he said the car stalled, he was able to get it running again for about another 10 minutes until he stalled going up another overpass. This time he pulled over and stopped the car, it would crank but not turn over. Someone offered him help and gave my husband a jump and from there the car started again, and he drove home and almost when he made it to our drive way the car stalled again and he coasted in. At that point the car was cranking but still not turnning over, he went to sleep because he works over night and after 10 or so hours tried to turn the engine over again and nothing it will not even crank or try. He stated the car was making a ticking sound and he staed he thought it was a Lifter arm? However he also said it was a dead battery. I asked if the lights came on and he said yes I said the battery cannot be dead because the lights come on it does not make sense. So he is gone now and took the cars battery to the local auto zone to recharge the battery. Any help would be greatly appreciated. Thank you!