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I think you'll find that the appearance of the mother's maiden name in an online birth index in either Scotland or England is related to the 100-year rule if you are looking at the indexes on either ScotlandsPeople or the GRO. The only exception within the period covered by the 100-year rule that I have noted is that if you know the mother's actual maiden name and enter it as part of the search criteria, it will show the mother's maiden name in the search results if the birth is a post-1923 birth. This is certainly the case for my own children who were born in England in the late 1990's/early 2000's. I can find their birth registrations using my own surname, but without mother's maiden name mentioned in the search results, but if I specify their mother's maiden name as part of the search criteria, it does show their mother's maiden name in the search results.

ScotlandsPeople has over the last few years slowly been adding mother's maiden name to all their indexed births from 1855 onward, up to 1923. They are also adding them to the indexed deaths as well, although there is still a gap from the late 1920's up to 1974 when they started being routinely indexed.

And finally, given the apparent Warwickshire/Gloucestershire links, this looks to be the likeliest candidate for Fanny's birth in England.

ASTON, FANNY        Mother's maiden name: HUGHES     
GRO Reference: 1872  S Quarter in SHIPSTON ON STOUR  Volume 06D  Page 605

I can also find a sequence of three Aston/Hughes births registered in nearby Evesham that match the names and ages of the three older siblings of Fanny listed on the 1874 passenger manifest.

Edited to add that this appears to be Fanny's family as it was in the 1871 England and Wales census, living in Weston Subedge, Gloucestershire, more or less halfway between Evesham and Shipston on Stour.

https://www.familysearch.org/ark:/61903/1:1:V5Y7-J5Q

It looks as though Laurence had at least four other siblings born between 1905 and 1911, with mother's name also being recorded as Fannie on one occasion.

1905/5524   Moses    Gladys Fanny                    Fanny   William John   
1907/17308   Moses    Hedley William George    Fanny   William John   
1910/5196   Moses    Phillis Mildred                    Fannie   William John   
1911/23042   Moses    Winifred Mary                    Fanny   William John   
1914/19686   Moses    Laurence Bramwell            Fanny   William John   

If the first child of the marriage was born in 1905 it suggests the marriage took place within a couple of years of this date, which then leads to this marriage of a Fanny Aston and William John Moses in 1903.

1903/6727   Fanny    Aston    William John    Moses

A printout of the marriage registration would presumably contain more information about Fanny's parents.

William's 1936 obituary in the Ashburton Guardian states he was from Cornwell, and also names Fanny's father as George Aston.

https://paperspast.natlib.govt.nz/newspapers/AG19360602.2.18

Edited to add that Fanny's entry on Findagrave states that she was born in 1873 in Warwickshire, England, although the evidence for this statement is not given.

https://www.findagrave.com/memorial/206669412/fanny-moses

I suspect the answer to the first question would be a fairly firm no. Since the objective of the post-1920 renumbering was to give each soldier a unique service number that would remain with them no matter where they went, and prior to this regiments had their own individual numbering systems which meant that there were likely to be multiple soldiers with the same service number, it would have defeated the purpose of the post-1920 renumbering if they had retained their old service numbers.

As for your second question, the new service numbers were allocated in blocks of numbers, with each block being assigned to a particular regiment, see the Long Long Trail on the post-1920 renumbering.

https://www.longlongtrail.co.uk/soldiers/a-soldiers-life-1914-1918/new-british-army-numbers-issued-in-1920/

What I can't tell you is how each regiment then initially allocated the first set of numbers in their assigned block to those soldiers serving at the time. It could have been in date order according to when they enlisted, it could have been alphabetically, or indeed some other way. I think the two most logical ways are the ones I've mentioned above, doing it by rank and seniority would just be overly complicated. Certainly I've seen both ways mentioned above used when looking at the WW1 medal rolls and the renumbered 1917 Territorial Force service numbers. In fact the Long Long Trail has this to say of the 1917 renumbering of the Territorial Force, and how the new numbers were allocated:

The distribution of numbers to the different elements of a unit followed no set pattern.

https://www.longlongtrail.co.uk/soldiers/a-soldiers-life-1914-1918/renumbering-of-the-territorial-force-in-1917/

You can look it up in one of the editions of the Gazette, you'll notice you have the reference to the date it appeared in the London Gazette. It also appeared in the Edinburgh Gazette a couple of days later on 21 January 1919. Unfortunately it seems it was one of a number that were all awarded for distinguished service during the war, rather than a specific action or deed.

https://www.thegazette.co.uk/Edinburgh/issue/13390/page/414

And I have now found the same in the London Gazette of 17 January 1919.

https://www.thegazette.co.uk/London/issue/31130/supplement/885