Please cite/link your sources, if possible. At some point in the next few days, I will:
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71 is the number of chicks that, according to an ancient superstition mentioned in the Gemara (Shabbos 67b), were to be counted at once to prevent them from dying. The Gemara forbids this practice as "darkei ha'Emori," the pagan rites of the Amorites. (Say, that rhymes!)
Sanhedrin had 71 = 70 + Nasi. The head of the 70 is Av Beit din. See Rambam, Hilchot Sanhedrin 1:5 (http://kodesh.mikranet.org.il/i/e101.htm) Example mentions of Sanhedrin of 71: http://kodesh.mikranet.org.il/b/h/h44.htm#1.5
The Diopelestion building in Alexandria of Mitzrayim had 71 chairs of gold, corresponding to the 71 members of the Sanhedrin. (Succah 51b)
I'm making an assumption here, but:
The kerashim of the Mishkan were 1 1/2 amos wide and 1 amah deep. According to the braisa that holds that there were three middle rods, the nominal length that the rods held in place was 72 amos (30+30+12) [I have no proof, but I'm assuming the corners were held by the western rod, not the north and south ones].
However, according to Rabbi Yehuda (Shabbos 98b), the depth of the kerashim were gradually shaved from an amah to a point on top (k'etzba). In order to compensate for the jut at the two corner planks, these corner planks were shaven to meet the shape of the north and south walls- from 0 at the bottom to an amah at the top. The center would have a 1/2 amah shaved off on each side. So the middle rod covered an area of 30+30+11=71.
(Though it doesn't seem like the actual rods were that long, or they would have to be shaven, too, so it wouldn't be proud of the keresh, and that isn't mashma.)
Number of languages (e.g., Rashi, B'reshis 50:6).
71 middos of a perush. Chovot HaLevavot, Ninth Treatise, ch. 4.
