The Sages banned hot bathing because people were heating water on Shabbat and claiming it was pre-heated. Cold water rinsing remains permitted.
Question 2
Why did the Sages forbid looking in a metal mirror on Shabbat?
A metal mirror's shiny surface might tempt you to pluck a stray hair — which is a toldah of shearing. Glass mirrors (which can't be used for plucking) are permitted by later authorities.
Question 3
Can you hang a curtain on Shabbat?
The tent prohibition targets overhead covering. A curtain that hangs vertically (like a room divider) doesn't create a tent — it's a partition, which is permitted.
Question 4
Can you wear a wide-brimmed hat on Shabbat?
A hat with a brim that provides shade is considered personal clothing, not tent construction. You're wearing it, not building it.
Question 5
Your Shabbat garment has mud on it. How can you clean it?
Washing (laundering) and wringing are melachot. But rubbing from the inside — making the mud fall off without water — is permitted.