Ah, yes. This parent I've focused on more than any other parent in the series, given my numerous fanfics that focus on her family's history of struggles, from her mate's disappearance and her brother's misdeeds, to having to raise her children alone. In spite of these possibilities being among her hardships, she does such a fantastic job at raising Petrie and his siblings despite their antics, particularly Petrie himself, and she brings a somewhat serious but otherwise warm presence when she soars into the picture. IMO, Mama Swimmer is the warmest parent in the entire series, but Mama Flyer comes in a very close second (not counting Littlefoot's mother because she didn't even survive the first 30 minutes of the first film), given her warm, unconditional love for Petrie in spite of his early flying troubles and recurring disobedience, which is especially shown in the 12th film.
1. Being a single mother of several flyer children has to be a herculean task for Mama Flyer, especially when one considers the antics that Petrie and his friends get up to on a regular basis. What are your thoughts of her parenting style, based upon what we have seen in the films and the television series?
Disregard the fact that none of the Elders seem to be able to keep their children in line, and you might find that Mama Flyer is an extremely devoted mother. It may be difficult to look after multiple children, especially Petrie, but she loves each and every one of her children and would do anything to make them happy, such as when she tried in vain to put a smile on a self-loathing Petrie during his failed practice flights or when she willingly put herself in harm's way just to rescue Petrie's precious snuggling stick. Her love for her son only left us praising her for her ability to be a warm motherly figure, and left us begging for more given how little we see of her.
2. Do you think the fact that Petrie and his friends found the valley on their own has affected how Mama Flyer handles Petrie in relation to his non-flyer friends? For example, we do not see the same skepticism in relation to them as we see in Cera’s dad (though, in fairness, he is an extreme example).
We never got any glimpses into Petrie's possibly rough upbringing prior to the Earth Shake, but I can assume that Mama Flyer secretly had a good heart towards many plant-eating children during those painful years and wished she could do something to change the world. Littlefoot's mother, I'm sure, had that same mindset, and it's very possible that Ducky's mother did, too, given the warm kindness she gives to all her daughter's friends. She may find it difficult to trust Petrie's friends from time to time given how many times they sneak out of the valley, but given that these kids were responsible for forever changing Petrie's life for the better, she has a big heart towards all of them. I can't really say for sure how skeptical she may have been towards Chomper and Ruby being welcomed into the valley, but the fact that Chomper would later save her life sure cemented her faith in Petrie's judgment when it comes to making friends of all sorts.

3. There are many unanswered questions concerning Mama Flyer in the series, such as: “What happened to her mate?” and “Is this her first clutch of children, or has she had clutches in the past?”. Do you have your own preferred head-canon on those or other unanswered questions?
In my head-canon, her mate died just a day after the children hatched, trying in vain to find a safe new haven for the sake of the family's future. As such, I believe that Petrie and his siblings were Mama Flyer's only clutch (I gave Petrie four siblings in my fanfics). Henceforth, Pterano had to help her sister raise her children, which became quite a challenge when Petrie's cowardice and inability to fly brought the family shame and humiliation. Of course, that ended up being all for naught after Pterano's own fall from grace.

I'm sure we all have many unanswered questions about Mama Flyer, but those regarding her family's unfortunate past are just the tip of the iceberg.