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The one born in 1900 would have been 24 in most of 1925.

396795 Thomas Bee, born Laurieston, Edinburgh, aged 21 years 11 months, enlisted in the Royal Tank Corps on 14 November 1921 at Edinburgh. Previously 42650 Scottish Rifles, 1 year 131 days. Wife Dorothy. Married A'shot, child born 25.4.25. Discharged 13.11.33.

1. Would it be the case in early 1850s that an unmarried woman having a child would be mentioned in Kirk Session or only in cases where she was prepared to name the father?

A pregant unmarried woman who was a member of the Church of Scotland congregation would be summoned to the C of S Kirk Session and 'interrogated' to name the father. Mostly they did, but sometimes they refused.

If she did not belong to the C of S, for example if she was Roman Catholic, she would not be summoned before the C of S KS.

2. Would the 'hard copy' original of Kirk Session records be available elsewhere e.g. at National Records of Scotland, Mitchell Library for consultation? I can't find digitised Kirk Session records for 1850s Lesmahagow on Scotlands People website nor on Oldscottish.com

You can check what KS records are in the care of the National Records of Scotland by searching the NRS catalogue. I did so, and it seems that the Lesmahagow Kirk Session records for the 1850s have not survived. See screenshot.

The yellow dots mean that the documents that do exist have been digitised and the hard copies are no longer made available for research..

I am not aware of any surviving KS records being held anywhere other than the NRS, but it's not impossible that the odd one has escaped. In that case, either it will be catalogued in either the NRAS Online Register or the Scottish Archive Network, both of which can be reached from https://www.nrscotland.gov.uk/research/catalogues-and-indexes, or you are never going to find it.

3. Could an unmarried woman claim to be married on Census records, birth records?

Yes.

Would this be checked in any way?

Unless she was personally known to the enumerator, no.

McReady is spelt several different ways in various records which doesn't help with the search.

Use wildcards - search for M*c*r*d* to find all variants.

surname is transcribed on SP as 'McKeddie

Use the yellow error button at top right to notify SP of the error. They will check it and will correct it or possibly add it as an alternative spelling.

Funny you should say that, Lodger - I've never come across a Rutherglen forum or page on RootsChat either in the last 18½ years.

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According to the index at Scotland's People www.scotlandspeople.gov.uk Daniel McKinlay and Isabella Muirhead were married on 13 June 1830 in the parish of Barony, Lanarkshire. Barony is part of the city of Glasgow.

Also in the baptisms index at SP are three children of Daniel M*c*k*n*l and Isabella Muirhead. There's also one to Peter M*c*k*n*l* and Isabella Muirhead. It would not surprise me if Peter is an error and Izobel is another child of Daniel's. See screenshot.

Obviously they were still in Scotland in 1841, and I think they were in Shettleston, parish of Barony, in 1841. See screenshot from https://freecen1.freecen.org.uk/cgi/search.pl

Daniel is listed as aged 40 in 1841. Adults' ages were supposed to be rounded down to the nearest 5 years, and the census was taken on 7/8 June 1841, so someone recorded as aged 40 should have been born between 7 June 1796 and 6 Jun 1801. This fits the birth year you have for your Daniel.

There is a baptism of a Daniel McKinlay in the parish of Greenock, Renfrewshire, in 1797, but there are no surviving records of baptisms of Daniel McKinlays in Lanarkshire in 1797.

Going back to the 1841 census, there's a Janet who could be a daughter of either Daniel or Isabella by a previous relationship, or their own daughter born before their marriage. Then there are
Daniel, born 1831/1832 (from census)
William, born 1834/1835 (baptism and census)
Izobel 1837 (baptism) - probably died in infancy
Peter 1840 (baptism and census)
Isabella 1844 (baptism)

They must have emigrated between 1844 and 1851 - I have not found them in the 1851 census.