Random letters for word games, classroom drills and drawing prompts: one at a time, a whole row, or a column you can work down.
Every letter here is equally likely, which is not how English works. In real text E appears around 12 percent of the time and Z around a tenth of a percent. That difference matters depending on what you are doing. For a fair game round (pick a letter, name something beginning with it) uniform is exactly right, because weighting toward E would make the game repetitive. For anything meant to resemble English text, uniform letters look conspicuously wrong: too many Qs and Zs, not enough vowels. The vowels-only mode exists for the specific case of filling vowel slots in a word puzzle.
Uppercase is the default because a single letter is more legible capitalised, and because most letter games are written that way on a board. Mixed case is for password-style drills and typing practice where shift usage is part of the exercise. The output options change what the result is for: spaced letters are readable at a glance for a game round, one per line suits a worksheet column, and joined letters produce a continuous string of the kind you would use as a test fixture or a nonsense token.
Yes. That is deliberate for game fairness, though it does not resemble English, where E is roughly a hundred times more common than Z.
Not individually. The vowels-only mode is the one built-in restriction; for anything narrower, generate a longer list and filter it.
Yes. Each draw is independent, so a short list can easily contain the same letter twice.