When it comes to writing down, first thing I do is to develop the character a bit. If s/he is going to have a place in my story, I need to know how they’re going to fit into it or at least have a good idea of what makes them so special. This usually only takes a few minutes, maybe hours, and then I write the idea down. Although, if the idea sticks around for a couple of days and nothing happens but I still feel like I’ll like the character, I jot down a few notes and take it from there. Sometimes it’ll become a full character, other times the ideas will be torn apart to help develop or reshape other characters.
Also, while I do admit (bad thing) I keep the ideas bouncing around in my head, I like to write as much down as possible. There’s only so much I can remember, and even if I never use it, I think it’s always nice to have a character profile in case I forgot something or I end up with inconsistencies. Also, I’m not sure what everyone here means by “profile;” it’s something I make myself on a blank sheet of paper. I write down as many important characteristics as I can think of—name, age, appearance, personality, weapons and fighting style, friends/relatives, rank (if applicable), race, faction, backstory, special powers, (possibly) the clothes they usually wear, and (possibly) role in the story, etc. I don’t do questionnaires or anything else static or pre-written; I only create characters if I get inspiration or if I feel like they’re necessary. Then I flesh them out as I feel is necessary and fit the story; questionnaires are inevitably generic and aren’t tailored to your specific story. If I’m not sure about a character I ask people, mostly my writer friends, and only go to a questionnaire if I’m still worried the character needs improvement.
Mary Sue? Long list of questions to ask myself. I can post it if you guys like, otherwise I'll try and keep this short.
Developing the characters changes a lot depending on what inspired them and how much of the story has been developed. For example, if they’re supporting protagonists their beliefs will have to be compatible with that of the MC, even if only reluctantly. Other times it’s just a concept—a quote, a personality quirk, a weird power or corruption, inspiration I randomly get creates a character idea that I’ll somehow tailor to the story, etc. Or maybe it’s a role—I create a character to fill a role and slowly the character grows to become a deep, complete character. Of course, characters constantly change as the story progresses and the story becomes more complex. For example, a villain I had for a story was going to be an only child, but later I came up with ideas for siblings and it completely changed his personality. Or maybe I create a character that's has borderline morals but find that they work well in a very ruthless setting and thus they’re molded by it and become entirely ruthless themselves. Other times the setting may be based around the characters instead. Sometimes one character is all I need to shake up the entire cast, giving the villain new opportunities and creating tension with the MC (or vice versa).
Overall I like characters as developed as possible, but worst case I can throw in loosely-made characters and make notes to improve or expand on certain areas (most often appearance or name).