From 1 Timothy 3:15:
.(if I am delayed, you will know how people ought to conduct themselves in God’s household, which is the church of the living God, the pillar and foundation of the truth.)
Now, assuming that "the pillar and foundation of the truth" refers to "God’s household", what, then, was God's household at the time that Paul wrote?
Actually, a pragmatic reading of the verse makes synonymous all three items, God's household, the church of the living God, and the pillar and foundation of truth. They can't be extracted one from another.
Was it the universal Church, consisting of the Apostles, congregational elders and the rest of the believers in Christ, or was it a particular religious institution that didn't have its beginning until the year 1054 CE?
You're confused. It was the Eastern Orthodox Church that began in 1054, not the Catholic Church. The Holy See was, in all previous centuries going back to 200 A.D. the Synod of Rome. It wasn't until 1054 that the Synod of Constantinople broke union with the rest of the church.
Was "God's household" the universal Body of Christ consisting of all believers, or was it a particular group of believers who became a separate unit later in time, a unit headquartered in Rome?
The Church cannot be separated from its leadership, the 12 apostles ("did I not choose you twelve?") who were directly commissioned by Christ and the bishops and presbyters they appointed starting a long line of apostolic succession. The Church is the whole body of Christians not excluding the authority structure in place right from the beginning.
Answer: It was the universal Body of Christ consisting of all believer.
Today, the universal Body of Christ consists of believers in the Messiah who are messianic Jews, Protestants, Roman Catholics, Eastern Orthodox and Coptic.
Incorrect. The Catholic Church is not one of many denominations, it is the original apostolic church that Jesus started, of whom he prophesied, "the gates of hell shall not prevail against". The Eastern Orthodox Church also lays a valid claim to apostolic succession, having bishops that directly descend from the original apostles like the Roman Catholic church. All others outside the apostolic church are separated brethren of Christian like leanings, but not in full fellowship with the original, enduring, universal church that Jesus himself established.