dy/dx

Q1.
y' = [(1 + e^t)d(t + sin t) - (t + sin t) d(1 + e^t)]/(1 + e^t)^2
= [(1 + e^t)(1 + cos t) - (t + sin t)(e^t)]/(1 + e^t)^2
= [(1 + cos t) + e^t + e^t cos t - t e^t - e^t sin t]/( 1 + e^t)^2
= [ 1 + cos t + e^t( 1 + cos t - sin t - t)]/(1 + e^t)^2.
Note : e^t is missing in your answer.
Q2. is correct.