Question

Comparing cooling temperature changes at different stages of the universe's life provides evidence of
the rate of the universe's initial expansion
. Measuring the universe's ratio of hydrogen and helium provides evidence that these elements were
formed long after the universe had cooled
.

Answer 1

I’ll mark each statement and give a short explanation/correction.

1) "Comparing cooling temperature changes at different stages of the universe's life provides evidence of the rate of the universe's initial expansion." — True (with clarification).
- The cosmic microwave background (CMB) temperature and its redshift evolution (T ∝ 1 + z) and the thermal history inferred from e.g. CMB anisotropies, spectral distortions, and molecular excitation at high redshift, all constrain the expansion history and early expansion rate. Measurements of the temperature and its changes at different epochs therefore provide evidence about how fast the universe was expanding in those epochs.

2) "Measuring the universe's ratio of hydrogen and helium provides evidence that these elements were formed long after the universe had cooled." — False.
- The observed H/He ratio (≈75% H, ≈25% He by mass) is a primary piece of evidence for Big Bang nucleosynthesis (BBN), which occurred within the first few minutes after the Big Bang when the universe was still extremely hot. Helium and the other light elements were produced early, not long after the universe had cooled; their abundances reflect conditions (temperature, density, expansion rate) during those first minutes.