The simple understanding is that Chazal instituted the mitzvah to light candles or oil on Chanukah in order to commemorate the miracle that happened to Klal Yisroel after they won the war against Antiochus. After Klal Yisroel consecrated the Bais HaMikdash and went to light the Menorah, Bnei Yisroel found enough oil to light the Menorah for one day. The single flask of oil burned for eight days, until more pure oil was able to be produced. To commemorate this miracle Chazal instituted to light candles for eight days.
If this is the reason for the mitzvah of lighting Chanukah candles, Zera Shimshon has difficulty understanding four halachos. 1. The mitzvah of Ner Chanukah is to light at sunset. (Shabbos 21b) 2. The mitzvah of Ner Chanukah is for each family to light one candle. (Shabbos 21b) 3. The Mitzvah of Ner Chanukah is to place in the doorway that faces the public street. If he lives in an attic... (Shabbos 21b) 4. The Halacha is to put the Chanukah menorah on the left side of the door so the Ner Chanukah is on the left and the mezuzah on the right. (Shabbos 22a)
On these four halachos the Zera Shimshon asks; firstly, the daily mitzvah to light the Menorah in the Bais HaMikdash was in the late afternoon (according to the Rambam it was also in the morning) and it was Assur to light it at night. This being so, it would have been more appropriate to institute the lighting in the late afternoon, at the time the Kohanim lit every day and not after sunset when it was assur to light.
Secondly, the Kohanim only lit one menorah for all of Klal Yisroel. Therefore it would be more appropriate to light in Shul for the whole congregation, similar to the menorah in the Bais HaMikdash, and not each family in their own house.
Thirdly, why did Chazal institute to light in the doorway next to a public street. The menorah in the Bais HaMikdash was inside the haichal.
And fourthly, why did the Gemara mention the place of the mezuzah in connection with the place of the Chanukah menorah.
Zera Shimshon answers in light of the Medrash that explains that Antiochus wanted to darken the eyes of Klal Yisroel with his decrees to prohibit performing milah, keeping Shabbos, and Rosh Chodesh (according to the Rambam, all of the mitzvos).
Since these decrees are described as darkening the eyes of Yisroel then the salvation is giving light to Yisroel. The main reason for lighting the Ner Chanukah is to remember the miracle of the oil burning for eight days and also to commemorate that we went miraculously from the darkness of not being able to do mitzvos to the light of doing mitzvos.
According to this we are now able to answer all the questions we asked.
We light the Chanukah menorah in the evening and not in the late afternoon because Shlomo HaMelech compares mitzvos to light. Chazal teach us that a candle in the day is not noticeable since there is sunlight; therefore we light only in the evening so the light of the candles will be noticeable, similar to mitzvos.
The main place people do mitzvos is in the privacy of their home. Therefore, Chazal instituted lighting in our homes. Since a mitzvah done in public has a special status, we also light Neiros Chanukah in Shul.
We light in the doorway to remember how Antiochus sent his representatives to check on Klal Yisroel to make sure they were not keeping the mitzvos.
Even nowadays, when Baruch Hashem, in most places in the world there are no decrees against mitzvah observance, we can still benefit from putting the Chanukah menorah in the doorway.
The Arizal explains, the mezuzah protects us from the yetzer hora on the left side. Chazal instituted to put the menorah opposite the mezuzah so we will be protected from the yetzer hora on the right, giving us protection on both sides. It is for this reason that Chazal mentioned the place of the mezuzah in connection with the place of the Chanukah menorah.
