Is it permissible to perform tayammum instead of ghusl to cleanse from janabah due to the cold weather, scarcity of available gas, and insufficient privacy in shared accommodation? And is it permissible to engage in intercourse after the cessation of menstruation but before performing ghusl?

If heating water is difficult due to a shortage or scarcity of gas, and there is no other way to warm the water, and there is a fear that using cold water will cause illness or worsen an existing one, then it is permissible to perform tayammum instead of ghusl, but wudu is not sufficient.

As for merely feeling shy about performing ghusl in front of people, while it is possible to do so, this is not a valid reason for tayammum.

Regarding intercourse after a wife's menstruation has ceased but before she has performed ghusl, it is not permissible according to the majority of scholars if the wife is able to perform ghusl. However, if she is unable to perform ghusl, then intercourse is permissible after she performs tayammum.