When will perihelion coincide with the June solstice? When this happens, will the average global temperature rise, fall, or stay the same?
Earth's perihelion will coincide with the June solstice in about 12,000 years. Northern hemisphere Summers may get a bit warmer.
Earth currently is at perihelion around 3rd January. It actually is at perihelion between 2 and 4 January due to the gravitational pull of the other planets.
The perihelion is precessing at a rate of about 5038 arc seconds every Julian century (=36524 years). This means that perihelion gets a day later every 70 years. Hence it will take about 12,000 years for perihelion to fall on the June solstice.
At perihelion the Earth is about 147,100,000 km from the Sun and at aphelion it is about 152,100,000 km. The difference is about 5,000,000 km or just over 3%. The southern hemisphere currently gets stronger sunlight in its Summer due to being close to perihelion. The situation will reverse when perihelion is at the June solstice. As the northern hemisphere has a lot more land area it will get a bit warmer in the Summer.