Question

When the mass of the sun is larger, Earth moves around the sun at a faster or slower pace.

When the mass of the sun is smaller, Earth moves around the sun at a faster or slower pace.

When Earth is closer to the sun, its orbit becomes faster or slower.

When Earth is farther to the sun, its orbit becomes faster or slower.

Answer 1

-- The mass of the sun never increases.

-- It does decrease, but not nearly enough to have any noticeable
effect on the orbital motion of the Earth, or any other planet.

-- When Earth is closer to the sun, it moves faster in its orbit.

-- When Earth is farther from the sun, it moves slower in its orbit.

-- The result is that the line from the sun to the Earth always covers
the same amount of area in the same length of time. 

-- Johannes Kepler noticed this, and it's his Second Law of planetary motion.

-- Newton showed that if his equations for gravity and motion are correct, 
then planets MUST behave this way.