Dought & clear, - sacrifice, - His sacrificial animal fell and he slaughtered it before it died; does it still count as a sacrifice?.

The sacrificial animal fell from the roof of our house and we
slaughtered it before it died; is that permissible?.
Praise be to Allaah.
Firstly:
It appears from your question that you slaughtered the sacrificial
animal before the Eid prayer. If that is the case, then it is not a
sacrifice (udhiyah), because the condition of the sacrifice is that it
should be slaughtered on the days of sacrifice, which are the day of
Eid and the three following days.
It was narrated that Jundub ibn Sufyaan (may Allaah be pleased with
him) said: I was present on Eid al-Adha with the Messenger of
Allaah(peace and blessings of Allaah be upon him). When he had
finished leading the people in prayer, he looked at a sheep that had
been slaughtered and said, 'Whoever slaughtered (his sacrifice) before
the prayer, let him slaughter a sheep in place of it, and whoever has
not yet slaughtered (his sacrifice), let him do so in the name of
Allaah." Narrated by al-Bukhaari, 942; Muslim, 1960.
Based on this, if this sacrifice was in fulfilment of a vow, then you
have to offer another sacrifice in its stead.
But if you slaughtered it at the time of sacrifice, and you bought it
with the intention of sacrificing it, then it is valid and is a
sacrifice (udhiyah), even if it was damaged as the result of its fall
from the roof. See also question no. ( 39191).
Secondly:
With regard to the validity of your slaughtering it, it is valid if
you managed to do it before it died.
Allaah has forbidden that which has been killed by strangling, and
that which has been killed by a violent blow - which is an animal that
dies from a blow with a piece of wood or metal - and that which has
been killed by a headlong fall - like your sacrifice - and that which
has been partially eaten by wild animals. This applies if the animal
dies, but if people catch up with it before it dies and slaughter it
in the manner prescribed in sharee'ah, then it becomes halaal.
Allaah says (interpretation of the meaning):
"Forbidden to you (for food) are: Al-Maitah (the dead animals -- cattle
-- beast not slaughtered), blood, the flesh of swine, and that on which
Allaah's Name has not been mentioned while slaughtering (that which
has been slaughtered as a sacrifice for others than Allaah, or has
been slaughtered for idols) and that which has been killed by
strangling, or by a violent blow, or by a headlong fall, or by the
goring of horns -- and that which has been (partly) eaten by a wild
animal -- unless you are able to slaughter it (before its death)"
[al-Maa'idah 5:3]
Ibn Katheer said:
The phrase "unless you are able to slaughter it" refers to that which
you are able to reach and slaughter in the proper manner when there is
still evidently life in it. That refers to the phrase, "that which has
been killed by strangling, or by a violent blow, or by a headlong
fall, or by the goring of horns -- and that which has been (partly)
eaten by a wild animal".
Tafseer Ibn Katheer, 2/11, 12
It was narrated from Ka'b ibn Maalik (may Allaah be pleased with him)
that they had sheep that used to graze at Sal' (a mountain in
Madeenah): a girl of ours saw a sheep that was dying, so she broke a
stone and slaughtered it with it. He said to them: "Do not eat it
until I ask the Prophet(peace and blessings of Allaah be upon him)."
He asked the Prophet(peace and blessings of Allaah be upon him) about
that and he told him to eat it. Narrated by al-Bukhaari, 2181.
And Allaah knows best.