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.
Related questions
If the Moon is debilitated, is a North-West entrance bad?
Yes, not just bad but very bad; the North-West entrance will have to be closed.
If the Sun is debilitated (neech) in the horoscope, is an E4 door right?
No, in that case an E4 door is completely wrong.
If the main door is on a negative pad, does a clockwise or anti-clockwise route to the boundary gate change its results?
Yes. A correctly placed door with an anti-clockwise (heen bahu) exit gives fewer good results than it should. A door on a negative pad such as E5 or E6, whose route runs clockwise to a gate on W4 or W5, gives much less negative effect than the pad alone would.
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.
Is choosing the door related to astrology?
Yes. With the person's astrology as the base, the door is chosen according to whom it will suit. The class covers three ways, by rashi, by exalted and debilitated planets in the birth chart, and by mulank, and the birth chart carries the most weight.
Must the house be at 0 degrees for a sarvatobhadra dwar?
By definition it is at 0 degrees with doors in all four directions, but it can be made otherwise too: build the doors on whichever pads fall on Pushpdant, Grihakshat, Bhallat and E4.