why it is not so?

You have asked a very interesting question. You can read up more details on the url given below.

http://www-istp.gsfc.nasa.gov/stargaze/Kep3laws.htm

Let me answer the question that you have asked -

First of all when a planet is further away from the sun its velocity increases( note that it does not decrease), the closer it  is to the sun the faster it moves.

This can be proved by taking the area as a right angle triangle with its distance from the sun as the height of the triangle and the arc length covered as its base. This proof is also given in the url mentioned above.

The earth actually spends 2 days fewer in winter than in summer. You can actually look up the calendar and count the number of days from a spring equinox to autumn equinox(on these days the day and night are equal).