If the Sun is debilitated (neech) in the horoscope, is an E4 door right?
No, in that case an E4 door is completely wrong.
Related questions
Can doors be opened anywhere by vastu, or must the birth chart be checked?
Check the birth chart. If the Sun in the chart is debilitated and you forcibly open an E3 or E4 door, it will be wrong and not appropriate at all.
In the example of the Sun in the 8th house, would an East entrance have suited if the Sun had been retrograde?
Possibly, but then its degree and the aspects on it, such as a benefic planet's aspect, would have to be checked. A house is built only once, so such a risk should not be taken. Retrograde (vakri) planets are taught later in the course.
The kundli chart was shown with directions. How are those directions used, and what can a kundli tell about a house?
Go by the houses, not the numbers written in them: the top house is the 1st house (East) and the 4th house is North. If the planets on the North side are strong, the North direction is very good for the person. A kundli can even reveal where a person's house is, where its door is, how many windows it has and where a nearby river, pond or temple lies, so a map of the house can be drawn without seeing it.
A family with an N4 door has had only boys across generations. What can be done for a girl child?
N4, like N3 and W4, is a door that brings sons. Open an E2 door, which brings the birth of daughters. A home without a daughter is not a complete home.
Can a person whose Sun is debilitated (neech) have the drawing room in the East?
Yes, with remedies; he will not leave the East unused or build a toilet there. A drawing room is not a permanent seat. Only permanent seating, such as an 8-hour business seat, should not be chosen in the zone of a debilitated planet.
For someone with no astrology background, what does an exalted or debilitated planet mean in practice?
Learn the twelve rashis in order and the rest is simple. If the Sun is written with rashi 1, it is exalted, very powerful, and blesses the house and direction it occupies. If the Sun sits with rashi 7, it is debilitated and very weak; if that falls in the 4th house (North), building a house in that direction ruins the person. Vastu without this is incomplete, which is why even a perfect East-facing house can fail.