Diddy,
Here is the extract from Merchant Tailor's School alumni records for Alexander:
And from Google Books:
Merchant Taylors’ School (London, England). A register of the scholars admitted into Merchant Taylors’ School, from A.D. 1562 to 1874, compiled from authentic sources and ed. with biographical notices, by the Rev. Charles J[ohn] Robinson, M.A., Rector of West Hackney. vol. 1, 1882. Lewes: Printed and published for the editor by Farncombe & Co., 1882-1883.
p. 91: “Alexander Culmer, b. 1 Nov., 1604.”
I have Alexander 1604 as the probable younger son of Richard Culmer c1571 - 1646 and his wife Cecily Sprackling Pre 1580 - 1623 St Peters in Thanet, with siblings:
Richard c1597 - 1662 aka the infamous cleric "Blue Dick". See entry in DNB.
Elizabeth 1599
Jane 1602, who married Leonard Rowentree c1581-1624? vicar of St Peters in Thanet
The Alexander 6/2/1598 - 27/11/1598 was the only child of John Culmer and his first wife Alice Norwood, who died in 1599. The family you mention (I'm 95% sure but lack absolute written proof, that I descend from George 1615) is by John's 2nd wife Margaret Sampson.