How likely are you to have a card in hand by a given turn? The answer depends on the game: opening hand size, whether the first player draws and how many cards you see each turn all differ. Pick your game and the calculator uses its rules.
These pages work from numbers you type in. In the deck builder the calculator knows your actual list: click any stat, such as your land count or your two-drops, or pick cards from your deck, and each becomes a condition, up to four at once. Cards that never come from your draw deck are left out for you, so the odds match the deck you will really play.
Drawing from a deck is drawing without replacement: every card you see changes the odds of the next one. The maths for that is the hypergeometric distribution, and it is exact, not an estimate.
A generic hypergeometric calculator only knows deck size, copies and cards drawn. The per-game calculators work out how many cards you have seen by each turn for you, going first or second, so you only enter your deck.
It gives the exact odds of drawing a number of copies of a card from a deck when cards are not put back. Deck size, copies in the deck and cards drawn are all it needs.
Games deal different opening hands and draw different numbers of cards each turn, and some skip the first draw for the player going first. Each calculator counts the cards you have seen using its own game rules.
No. The odds are for the hand you are dealt and the cards you draw after it, before any mulligan.