In general, no, the time does not change as you cross the equator.
solstice
No, the time does not change as you move north or south along the same longitude line. Time zones are determined by lines of longitude, so as long as you stay on the same line, the time remains constant.
It depends how far north, or south, of the equator you live.
At the equinox (either one of them) the Sun is directly above the equator, neither north nor south.
Moving across the equator, or moving north and south in general, does not alter the time or the date. It is by moving east and west that you can alter the time and date. The most abrupt change comes at the International Date Line.
Summer occurs south of the equator at the same time winter occurs north of the equator and vice versa.
Nov-Dec-Jan-Feb. Since most of South America is south of the equator, seasons are reversed from North of the equator. At the equator, it is warm all year.
Summer and winter
The Tropic of Capricorn is located South of the equator, although it is not in a fixed location; it varies over time.
The sun is farthest north of the Equator during the summer solstice, which occurs around June 21st each year, and farthest south during the winter solstice around December 21st.
Time zones change from east to west, so the correct answer is no. However they are not perfectly straight, so it is possible to be in a different time zone than a place north or south of you.
Areas farther north or south of the equator receive less sunlight because the angle of the sun's rays is lower, spreading out the energy over a wider area. This results in cooler temperatures and shorter days in those regions, especially during winter months.