As to the Temples' dates, my sources show that the First Temple was dedicated 962 BCE after 7 years of construction, and was destroyed (finally in its last part) at the end of July in 586 BCE, so it stood 376 years in operation.
The Second Temple was placed in operation early in the year 516 BCE, some say at Hanukkah of 517, and with several major renovations, rebuildings, and rededications especially that of Herod circa 19 BCE...until its fall early August of 70 CE: thus it stood 586 years to nearest whole year.
Of course the notorious false messiah, bar-Kokhba, did rebuild and attempt to reinstitute the Temple services with the help of Rabbi Akiva for some 2 to 3 years circa CE 132-135, but this is now discounted and not considered a Third Temple.
Broadly, the present State of Israel represents in bud form, a Third Temple, and we all hope the actual Bet hMikdosh will be speedily rebuilt in our own days. That will require the ole Dome to get out of there, and good riddance.