Correct me if I'm wrong

In the first movie, we are told that Mrs Voorhees was working in the kitchen while counselors were supposed to be watching her son. The counselors were busy having sex, and her hydro-cephalic child went late night swimming and drowned. This implies a body was found, since no one would've been watching him. In Part 2, we are told by Paul that Jason's body was never recovered from the lake when he drowned.

#1) The only way people would know he drowned is if his body was found floating on the lake. But no body was found. So how do they know he drowned? Were there others around when he drowned? Were these people just listening to him flailing in the water and no one came to help him?

#2) If no body was recovered from the lake, then wouldn't that mean he didn't drown? Crystal Lake is specifically for the summer camp. In later films we find out it eventually condenses into a river that leads out to the Atlantic Ocean, but unless there was stormy weather, which is never hinted or implied, the police would have no reason to suspect his body washed into the river or anything. So that would mean if there was a body he'd still be in the lake. But no body was found. So rather than the official police report reading he drowned, wouldn't Jason Voorhees be declared MISSING? IMPLYING HE'S POSSIBLY ALIVE, and DEFINITELY NOT in the lake?

#3) If no body was found. how is the camp still open a year later? I'm pretty sure not being able to recover a dead body OUT OF A LAKE FOR KIDS TO SWIM IN means bad business for the Christies. Or was it supposed to be implied that NO ONE other than Mrs Voorhees tried to find her son's body?

#4) If no body was found, the police would have had to search the woods. But in this movie, Jason's been living out in the woods for 27 to 28 years. Are the police this incompetent?