It could be Hawkman. Given their history and connection in the DC universe, it would be a likely choice. Their relationship has always been complex and a pregnancy could add another layer to it.
The Flash might initially be in denial. He's used to Batman always having a plan and his death would be a huge shock. He would then turn his speed into a tool for investigation, trying to find out exactly what happened. Green Lantern could feel a sense of loss but also be spurred into action, using his ring to try and prevent future tragedies like Batman's death. He might create constructs in Batman's memory to inspire the team.
One idea could be exploring how the other Justice League members react to Hawkgirl's pregnancy. Some might be overprotective, while others could be worried about how it will affect her in battles. Another could be Hawkgirl dealing with her own fears and excitement about becoming a mother while still being a superhero.
There could be a fanfiction where Hawkgirl has a romantic relationship within the Young Justice group. For example, a story about her and a male character like Robin developing feelings for each other while on a mission together. This would add an interesting dynamic to the team and her character.
John Stewart's age can vary depending on the specific storyline and continuity in DC Comics. Generally, though, he's often depicted as being in his 30s or 40s.
I don't think Vixen dated John Stewart in the comics. The comic plots didn't show such a romantic connection between them. Maybe in some alternate universes or fan-made stories, but not in the mainstream comic canon.
Fanfiction.net is also a popular place. There are many different categories and filters you can use to narrow down your search to specifically Young Justice and Hawkgirl related fanfictions. However, the quality may vary, so you might need to read a few to find the really good ones.
Definitely not. Justice League is a creation of DC Comics. Marvel Comics has its own set of superhero teams and characters that are distinct from those in the Justice League.
One of the top stories could be 'Tower of Babel' which shows the vulnerabilities within the Justice League when their secrets are used against them. Another might be 'The Darkseid War' that involves an epic battle against Darkseid. And 'Crisis on Infinite Earths' also has a significant impact on the Justice League's story as it reshapes the multiverse they exist in.