Looking at the parish register for Broomfield, available (via subscription) at Essex Archives Online, there is the following entry:
Christofer Archer was married the 25th September 1575
His wife's name is not recorded.
If this is the right Christopher, it could mean Gabriel was born in, say, 1576, which would make him about 15 years old when he went to Cambridge and 16 or 17 when he was admitted to Gray's Inn, which seems reasonable.
There are apparently no baptisms for any children of this Christopher in Broomfield or surrounding parishes, which could suggest that Broomfield was his wife's home parish and that after marrying they lived in Christopher's parish (ie Mountnessing).
However, there is this burial entry in the Broomfield parish register:
Marie Archer the daughter of Christofer Archer was buried on a Moonday beinge the xxijth of September 1595
Christopher was dead by then, so perhaps his widow had moved back home. Or perhaps daughter Marie was living with her grandparents or other relatives at Broomfield when she died.
While it's hard to be certain, based on the above it seems plausible that the Christopher Archer who married in 1575 at Broomfield is the same person as Christopher Archer, father of Gabriel.
