Roosters are not necessary for hens to lay eggs. They ARE necessary if you want to hatch chicks from the eggs.
That fella in that last photo is definitely your rooster. He's an EE cockerel (young rooster).
Your chickens are really good looking and a great mix of good layers!
Now, my dominant rooster is an EE and he's wonderful. I would never have a flock without a rooster, now that I have seen the benefits - and they're not just making eggs fertile for hatching.
If the crowing is a problem for your neighbors, I can understand you worrying about it. I did, too, but I give fresh eggs to my closest neighbors to bribe 'em. They lie and swear the rooster's crows don't bother them at all.
When hens lay eggs, they can make a WHOLE lot of noise, which could be just as obnoxious to neighbors, so consider keeping your rooster, if you can. I mean, since the hens will do the Egg Song loudly every time they lay, and every time another hens lays an egg they'll start up again, cheering her on.... just about as noisy as a rooster crowing.
It's just that a rooster's crow is so identifiable and easy to latch onto as something to complain about. If it's legal for you to have roosters where you live, you should keep one.
Just my opinion.