Why do I need my exact birth time to calculate my rising sign?
Because the Ascendant is the point on the ecliptic crossing the eastern horizon at that instant, and the horizon rotates 15° per hour; without an accurate time, there is no instant to calculate from, so the result cannot be trusted.
Does the rising sign differ even for two people born on the same day?
Yes, and that is the rule, not the exception: two people born on the same day but in different cities or at different times almost always end up with different rising signs, because the calculation depends on time and longitude, not just the date.
What is the Midheaven, and does it change along with the Ascendant?
It is the point on the ecliptic aligned with the local meridian at the moment of birth, tied to vocation in the traditional reading; it is calculated from the same local sidereal time as the Ascendant, so it also shifts with time errors, though at a different rate because of latitude.
Without a birth certificate showing the time, can the rising sign be estimated?
Only roughly, by testing likely times and seeing where the Ascendant crosses into a new sign: since the shift is about 1° every 4 minutes, a vague window like "morning" or "afternoon" can still cover more than one possible rising sign.