1) what age do pullets start being fertile? Or is it that their very first egg is the first chance at having an embryo
As soon as they start laying eggs they should provide their contribution to the DNA. The egg-laying process is pretty complicated, there are lots of chances for something to go wrong. That's why it is not unusual for the first eggs pullets lay to look weird. There are things you can't see that are required for an egg to develop and hatch that you can't see. To me it's surprising how many get all that right from the start.
You see recommendations to not hatch pullet eggs. There are valid reasons for that depending on your goals. If you are evaluating a hen for egg laying to see if you want to breed her you need to wait until she's laid a while. If you are breeding them for show you want to wait until they grow up before you decide which are your best.
I hatch chicks for meat and to play with genetics. I hatch pullet eggs. That fits my goals. I select my males for breeding when they are the age I want to eat them. I hold off on selecting which pullets are kept for my laying/breeding flock until I've seen them lay for a while. But I eat the others, nothing wrong with them.
I don't get as good of a hatch rate with pullet eggs as I do with eggs from hens that are older. Sometimes my hatch rate with pullet eggs is really good, sometimes pretty close to awful. It sometimes takes a while for a pullet to work out the kinks in her internal egg making factory. Practically every chick I hatch lives. When one dies odds are it hatched from a small pullet egg. Those eggs just aren't that big so they can't contain enough nutrients for a big chick so maybe they are a bit weaker. Or maybe something wasn't 100% right with the embryo even if it hatched. That can happen with eggs from older hens too but is more likely with pullets. I've found that if I wait until the pullet has been laying for a month before I set her eggs I'm a lot less likely to have these problems.
I think when to set eggs depends a lot on your goals.
2) when do cockerels reach fertility?
I don't know. Some will start trying to mate with pullets really young, some not so young. Some hens or pullets will let a young cockerel mate them, others won't. The girls have a part to play in that too.
Then there is the question of when their sperm is viable. I think that depends on how fast they mature. I butcher cockerels. When I open them up I can see that some appear sexually mature much younger than others. Some have testicles the size of a mature rooster while others are tremendously smaller. The ones more sexually mature are more likely to be accepted by the hens and pullets, so if they are letting him mate I'd assume he is capable of fertilizing the eggs.
3) Do male's lose fertility as they age?
I agree, they can. It might be physical fertility, the sperm just isn't as potent. But often it's vigor. They just don't mate with the hens like they used to. This depends on the rooster. Some five year olds don't have a problem keeping 25 to 30 hens fertile, some may not manage more than 3 or 4.
4) If you keep a male alone when he's going through hormones, can that make him not know how to breed anymore?
No, it's instinctive. Different chickens have different levels of instincts or hormones. Some hens go broody, some don't. Some broody hens instinctively keep their chicks really close, others let them roam further. Each chicken is different with their own personality.
Anything is possible, you may have a cockerel with messed up instincts, but I consider that highly unlikely. A 4 month old cockerel is still a young boy, maybe a pre-teen. He almost certainly hasn't matured enough for the hens or even pullets to willingly allowed him to breed. And the mature rooster will suppress his behaviors even if he wants to. He hasn't forgotten how, he just isn't old enough yet. Though often by 4 months he will be trying.
