Why do sign dates vary by a day between some sites?
Because the Sun does not enter each sign at the same instant every year, the tropical year has 365.24 days, so the exact turnover date can shift by 1 day depending on the year and the time zone used in the calculation; the range used here is the most common convention in Western astrology.
Does a zodiac sign match the constellation with the same name?
Not anymore: because of the precession of the equinoxes, the tropical zodiac (used here) has already drifted about 24° from the actual constellations since antiquity; someone born "in Aries" by the calendar has the Sun, in practice, aligned with the constellation Pisces in the real sky.
What differs between a fire, earth, air and water sign?
It is a traditional temperament classification: fire (Aries, Leo, Sagittarius) is tied to initiative, earth (Taurus, Virgo, Capricorn) to practicality, air (Gemini, Libra, Aquarius) to communication and water (Cancer, Scorpio, Pisces) to emotion; each element has exactly 3 signs.
Does being born on the cusp mean having two signs at once?
No: the Sun sign is always exactly one, defined by the exact birth date; "cusp" is an informal reading for people born within the 2 closest days of the boundary, blending a trait from the neighboring sign into the interpretation, without changing the official calculation.