No! In the case of the Philippian jailor, all those in his house were taught the gospel and believed the gospel (Acts 16:31-34); so in agreement with our Lord’s commission, the salvation offered by Paul to his house was secured by believing first and then being baptized (Mark 16:16).
No! In the case of Lydia, there is no evidence the woman was even married or had any infant children (Acts 16:14-15), for “household” can easily refer to domestic servants or other employees in distinction to children (Gen 18:19; Acts 10:7; Matt 10:25).
See Also
Infant Baptism?
Does circumcision prove infant baptism? No! Scripture teaches plainly and repeatedly that faith and repentance are conditions for baptism (Mark 16:16; Acts 8:37).