As I said in my first blog, I will put the updates on future episodes of Flame Princess' developments here. If I did not contribute, that means that I had either stopped watching Adventure Time, or lost interest in doing this, or being lazy/forgetful, or is unavailable in this wiki.
She is a short-tempered and aggressive mood swinger.
Hot to the Touch
She is quick to judge, but rather naive and curious, indicating that she had been locked up for a long time.
Not much for the character, as the episode is centered on romance, but it was shown that she is unstable to extreme romance and that she was locked up by Princess Bubblegum.
She wasn't given enough screen time to have a decent amount of development, but it shows that she had a strained relationship with her father and that she can be changed from evil to good.
Vault of Bones
After Flame King started harassing her with evil whispers, she developed a fear of being evil and desires to be good. Through out the episode, Finn guided her through the dungeon with his methods, which bores her immediately, showing her impatience and later admitting to Finn about it. Once she was given freedom, she goes mad with her powers, burning down doors and later fighting with the Goo Skulls. Her passion for burning things is most likely the fact that she is made out of fire, and as she said in "Hot to the Touch," fire's purpose is to burn. Later when Finn was captured by the Giant Goo Skull, she did not burn the Goo Skull immediately, as she knows that Finn would be burned as well, instead, she used Finn's method to defeat the Giant, showing her intelligence. This episode shows that Flame Princess is not good nor bad but Chaotic Neutral, she decides what is right or wrong and tries to avoid doing wrong stuff. By the end of the episode, she states that she will only destroy evil stuff.
She is shown to be spending more time with her less-evil family members as she prefers good to evil. She also prove that she is able to handle her anger well, as shown when she had calmed down after she figured out the bet despite the humiliation that Jake had forced Finn to do.
Frost & Fire
I am not very sure what is the development for her, so I will not add anything to this section here.
Earth & Water
This episode, I believe, have a huge amount of development for her. After her break-up with Finn, she started to believe strongly in truth and judges people by their honesty to ensure if they are a true friend. She dissent lies and wonders if everybody was hiding something after her discovery of Finn's betrayal and her past connection with Bubblegum and enacted a rule in the Fire Kingdom to have everybody be honest after overthrowing her father. Besides her personality, her backstory is also shown. A major flaw of Flame Princess, that she had shown in "Frost & Fire" and in later episodes ("The Red Throne" and "The Cooler") is that once a person, if he/she is likely to betray her, had gained her trust, she can be easily fooled and fall under their plans.
The Red Throne
Flame Princess shows that she is a caring ruler to her people, and works hard to help them with their problems.
This episode has shown that FP had matured from an innocent and naive girl to a wise ruler. She makes realistic solutions to her problems and holds onto the best options. Similar to "The Red Throne", she puts aside her feelings to call Bubblegum for help and agreed to open where the ancient relics of the Fire Kingdom in order to rescue her kingdom from the threat that they were dealing with. Despite this, she is shown to be still impulsive, destroying almost all of the Fire Giants just to get her revenge on Bubblegum. Her control of her anger seemed to have improved when she immediately forgave Bubblegum for what she had done to her kingdom after she promised Flame Princess that she is not a bad person.