Salaah

Heating the water for ghusl and performing fajr salaah in its time

Q: A person wakes up in need of a Fardh Ghusl 15 minutes before sunrise. There is no hot water and the person has chronic health issues. Using cold water will cause his health to deteriorate. It takes approximately 15 minutes to heat up sufficient water for Ghusl. If he waits, the time will expire and Fajr Salaah will become Qadha. What should be done in such a situation?

Masbooq starting his salaah while the imaam is in qa'dah or making salaam

Q: 

1. If the latecomer joins the Imam in the first or final tashahhud for even a moment, then does the latecomer finish his tashahhud before rising up (for either the third unit, or to complete his missed units)? If the Imam rises quite soon after the latecomer joins him in the first tashahhud, then is it a factor that the latecomer may be behind the Imam? 

2. A latecomer makes the opening takbeer while the Imam is in the first tashahhud. If the Imam begins to rise for the third unit while the latecomer is going down, what does the latecomer do? 

3. A latecomer makes the opening takbeer while the Imam is in his final tashahhud. If the Imam says the first Salam while the latecomer is going down to join him, what does the latecomer do?