A Torah work in doubt between the suns

Have a good week Rabbi,
I remember that it is permissible to perform a rabbinical work between the evenings for the purpose of a mitzvah, a great need, or to prevent a loss (metzah). Would it be permissible to perform a Torah work for the purpose of a mitzvah at the beginning of the evenings when there is doubt whether the time between the evenings has begun or not yet?
Additionally, is turning on the air conditioner on Hamsin Day considered a mitzvah or a great necessity?
Additionally, is pressing a button on an air conditioner remote with a fork considered a change? Or does it not matter how you press a button?