Is winter hotter at the Equator?
Does this really mean that on the Equator, where day and night are of the same length all year long, winter is the hottest season?


By "winter", I assume you mean December, January, and February. The answer is "No!"

Insolation varies on an annual basis outside of the tropics, one maximum and one minimum every year. This is not the case in tropical regions, where available insolation has two local maxima and two local minima every year. This effect is greatest at the equator. The graph below depicts available insolation at increments of 30 degrees latitude, from the equator to the north pole.



Note that at the equator, available insolation achieves local maxima at the two equinoxes and local minima at the two solstices. The reason is that the Sun is directly overhead at local noon on the equinoxes but is 23.5 degrees from vertical at local noon on the solstices. Sunlight has to travel through more air at the solstices than at the equinoxes.

The slight variations in insolation at the equator are easily overcome by climate. Equatorial regions tend to have wet seasons and dry seasons. When these occur depends much more on wind patterns than it does on aphelion / perihelion.




  Does this also mean that Winters in the Southern hemisphere are colder than our Summers and their Winters are hotter than or Summers?


The answer is once again "No". The slight variation in insolation due to the Earth's eccentric orbit is once again easily overcome by other phenomena. The driving characteristic for this part of the question is that the northern hemisphere has much more land mass than does the southern hemisphere. This means that except for polar regions, southern hemisphere seasons tend to be more moderate than do northern hemisphere seasons.