Do hens need a cockerel ?

I have 4 rhode island plus 3 "other" hens, + a cockerel, and they seem
to get on well together. They sleep in a tree by my bedroom.

The trouble is he wakes me up 5 am most days. Real bad for work, not
sleeping enough.

If I were to get rid of the cockerel, what adverse effects would that
have on egg laying , etc ?

Would my hens tend to go over to the neighbors garden and stay there ?
Lay their eggs there maybe ?

Also I have a broody hen, been that way for 40 days, she's been
sitting on an empty nest all that time. Tried dumping her in cold
water, putting her in a box for 3 days, etc etc, but she will not
"un-brood".
Ideas ?
.