Is Christ still bulding his church?

Yes, Jesus said in “And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.” (Matthew 16:18 KJV)

This he said before his resurrection. He had said he would build the temple of his body, which is his church in 3 days:

John 2:19-22 KJV Jesus answered and said unto them, Destroy this temple, and in three days I will raise it up. Then said the Jews, Forty and six years was this temple in building, and wilt thou rear it up in three days? But he spake of the temple of his body. When therefore he was risen from the dead, his disciples remembered that he had said this unto them; and they believed the scripture, and the word which Jesus had said. 

That was part of the allegation they charged him with at his trial and upon the cross (Matthew 26:61; Matthew 27:40; Mark 14:58). That he said he would build the temple (his church) in 3 days.

So, the question is, did he keep his promise? Did he build the church in 3 days? Or is he still building the church since then?

The Bible says believers are the temple of God and of the Holy Spirit (1 Corinthians 3:16-17; 1 Corinthians 6:16-20; 2 Corinthians 6:16-18).

Will God be living in a house that he is still building and that is uncompleted?

The Epistles, which are the letters written to the church after the resurrection of Jesus Christ, let us know that we are God’s building.

1 Corinthians 3:9 KJV For we are labourers together with God: ye are God’s husbandry, ye are God’s building.

Ephesians 2:20-22 KJV And are built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone; In whom all the building fitly framed together groweth unto an holy temple in the Lord: In whom ye also are builded together for an habitation of God through the Spirit.

Colossians 2:7 KJV Rooted and built up in him, and stablished in the faith, as ye have been taught, abounding therein with thanksgiving.

Hebrews 3:6 KJV But Christ as a son over his own house; whose house are we, if we hold fast the confidence and the rejoicing of the hope firm unto the end.

Hebrews 9:11 KJV But Christ being come an high priest of good things to come, by a greater and more perfect tabernacle, not made with hands, that is to say, not of this building;

1 Peter 2:5 KJV Ye also, as lively stones, are built up a spiritual house, an holy priesthood, to offer up spiritual sacrifices, acceptable to God by Jesus Christ.

Can we notice the tenses regarding this his house/temple/church?

When Jesus said the gate of hell shall not prevail against his building of his church, he was referring to the fact that death or the grave (hades) could not hold him down or stop the church. He was going to be raised and upon his resurrection, his church would be built. This the reason his disciples came to the realization that his statement about building his temple in 3 days upon his resurrection (John 2:22).

The church is not an ongoing project. Jesus has built his church in 3 days upon his death, burial, and resurrection. And because this building was not by human hands (Mark 14:58), it cannot be destroyed by man or any circumstance. Hallelujah.