There are ways to tell male and female newly hatched chicks apart, though usually it requires a professional as it can be a delicate procedure (I think its called like vent sexing or something like that). There are certain crosses that can be told apart based on their color, but I assume you are...
Signs a pullet is getting ready to start laying include: the comb/wattles getting redder, she will squat down when approached (this is what she would do for a rooster), she may look for and protect nesting sites, and she may get noisy.
The first egg is always exciting!
When I had sebrights they grew up fast! However, as they get older their combs and color tends to change some. They acted mature when they were really young, but it took them a while to fully grow into their behavior : )
I'm glad she seems to be doing better!
Long-term illnesses can lead to some permanent damage, so she may always be a little unsteady or weak. I had a young chicken who was apparently landed on by another chicken and it left her with brain damage. She tended to peck at nothing also and walked in...
They certainly could be Jersey Giants, especially if they were marked as such! Hatchery chickens don't always have the same size or exact coloring as chickens selected to meet the "standard", so this could be why they may look different from other Jersey Giants you've seen.