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Can anyone suggest what Isabelle's relationship and marital status in the attached extract are please?

Father was William McEwen not Alexander

Noted, but in that case where is Joan in 1851? And Alexander's occupation matches. Could there be an mistake in the marriage certificate? or in the census? Maybe Joan was Alexander's niece rather than his daughter.

Janet Hodge Paxton certainly points to Janet Hodge as Joan's mother. And Joan's age in 1851 fits her in the gap between James 1841 and Archibald 1846.

Alexander McEwan, mother's maiden surname Henderson, died in Inverkeithing in 1857, aged 53. His death certificate might be enlightening.

James Henderson, aged 69 died in Inverkeithing in 1881. Helen McEwan, 92, died in Inverkeithing in 1901.

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It does look to me as if he must have emigrated and/or changed his name.

Do you have any advice for what I could do to find him if he did indeed emigrate or change his name?

If he changed his name, no.

If he emigrated to NZ or some parts of Australia, there are death indexes online
New Zealand https://www.bdmhistoricalrecords.dia.govt.nz/Home/
New South Wales https://www.bdm.nsw.gov.au/Pages/family-history-research/family-history-research-nsw.aspx
Victoria https://online.justice.vic.gov.au/bdm/indexsearch.doj
Queensland https://www.qld.gov.au/law/births-deaths-marriages-and-divorces/family-history-research
South Australia https://www.genealogysa.org.au/resources/online-databases.html
West Australia https://www.bdm.justice.wa.gov.au/_apps/pioneersindex/default.aspx

but if he went to the USA and Canada and there's nothing in FamilySearch or on Anc*y, I don't know where else I'd look.

Their name changes to Gardiner in certain records.

It's not a change as such, just a variant spelling. Spelling was a very inexact science until into the 20th century. If you are searching on Scotland's People use the wildcard facility - g*rd*n*r should find all variants in one search.

Unfortunately that's where all records for him end, apart from the possible match in 1915 - he practically drops off the face of the earth. I can't find any reliable matches for deaths, marriages or anything via scotlandspeople.com, ancestry.co.uk or findmypast.co.uk.

https://www.rootschat.com/forum/index.php?topic=714261.0   

If you can't find it on Scotland's People you are wasting your time looking on any other web site for a death in Scotland.

I don't know if he re-enlisted after 1915 and was killed, or he emigrated or even went to prison.

If he stayed in England/Wales you could look for his death at https://www.freebmd.org.uk/search - same information as Anc*y and FindMyPast etc because they just mirror it, but I think it's easier to search.

My grandfather's father was his illegitimate son and John only appears on his birth record on scotlandspeople due to some kind of correction/addition, apparently due to the insistence of my great, great grandmother who knew who must have fathered her son.

The insistence of his mother, taken on its own, would have no effect whatsoever on the addition to the birth record.

There are only two circumstances in which a father's name appears on the birth certificate of an illegitimate child.

First, if the father accompanies the mother to the registrar's office when she goes to register the birth, and signs the certificate at the same time as she does.

Second, if a court of law orders an entry in the Register of Corrected Entries. This is usually following an action for paternity in the sheriff court, but there could be other reasons. You might be able to find the court records - the RCE should tell you which court and when, and the case might be indexed at https://www.scottishindexes.com/contact.aspx

It does look to me as if he must have emigrated and/or changed his name.