Is the third temple rebuilt before the rapture takes place?
The Bible never makes a connection between the timing of the resurrection (rapture) and the start of the Jerusalem Temple being built except that the resurrection must occur before the start of tribulation because scripture states that by the mid-point of tribulation the temple will be built and defiled by the Antichrist.
Additionally, tribulation begins with a deceitful peace treaty between the Antichrist and the Jewish people which could possibly initiate the build of the third temple. However, this is mere speculation. Marking the mid-point of tribulation the Antichrist will compromise the peace agreement, taking a seat in the temple himself and calling himself God, which in effect defiles the temple as described in 2 Thessalonians:
2TH. 2:1 Now we request you, brethren, with regard to the coming of our Lord Jesus Christ and our gathering together to Him,
2TH. 2:2 that you not be quickly shaken from your composure or be disturbed either by a spirit or a message or a letter as if from us, to the effect that the day of the Lord has come.
2TH. 2:3 Let no one in any way deceive you, for it will not come unless the apostasy comes first, and the man of lawlessness is revealed, the son of destruction,
2TH. 2:4 who opposes and exalts himself above every so-called god or object of worship, so that he takes his seat in the temple of God, displaying himself as being God.
2TH. 2:5 Do you not remember that while I was still with you, I was telling you these things?
2TH. 2:6 And you know what restrains him now, so that in his time he will be revealed.
2TH. 2:7 For the mystery of lawlessness is already at work; only he who now restrains will do so until he is taken out of the way.
2TH. 2:8 Then that lawless one will be revealed whom the Lord will slay with the breath of His mouth and bring to an end by the appearance of His coming;
2TH. 2:9 that is, the one whose coming is in accord with the activity of Satan, with all power and signs and false wonders,
2TH. 2:10 and with all the deception of wickedness for those who perish, because they did not receive the love of the truth so as to be saved.
The Antichrist will call himself god and will take a seat in the temple 3.5 years into the 7 years of tribulation (which must be rebuilt by the time of the Antichrist's appearing) and will rule the earth with the power of Satan until Christ's return. Furthermore, Paul says the specific identity of the Antichrist will not be revealed to the world until the "restrainer" is removed from the earth. The term "restrainer" refers to the Holy Spirit, Who lives in the world now (in the members of the Church), but He will leave the earth when the Church itself is removed at the resurrection (rapture). The resurrection of the Church takes place prior to the revealing of the Antichrist, therefore believers will not know the Antichrist's identity while we live on earth but may see the beginning steps to fortify the Jerusalem Temple. But again, this is not a prerequisite to the end-times timeline.
As we approach the end-times, we will continue to see increased preparations made to ensure the rebuild of the temple which started in the late 1980's, under the Temple Movement project. Furthermore, the Temple Mount Faithful, an organization that has been trying to prepare Israeli society to accept and promote the rebuilding of the temple, while making various Temple-related items, most notably a portable alter for sacrifice, are earnestly forging avenues to obtain proper worship at the Temple Mount.
As we eagerly wait for the arrival of the Messiah, the Church body eagerly awaits the rapture which can happen at any time.