You are very wise not to believe that tree. Never trust anything you find online unless it's an image of an original document, and especially do not trust trees submitted to Ancestry and similar commercial web sites. Use them as a basis for further research, checking the source of every detail.
And do not fall into the error of assuming that the only vaguely matching candidate in the surviving records must be the right one, because many people's baptisms were never recorded or, if they were, the record has not survived. Unfortunately too many people do make this error, and publish trees containing it, and other people come along and copy it into their own tree, and before you know it there are hundreds of trees with wrong information.
Bowden is a parish in the County of Roxburgh (aka Roxburghshire). I searched the indexes at www.scotlandspeople.gov.uk for baptisms ot T*m*s*n children in the parish of Bowden from 1745 to 1785. There are no recorded baptisms of children named Andrew, Euphemia or Robert, and none with father named Robert. I also searched FamilySearch and the IGI (separately) for a baptism of Thompsons, father Robert, anywhere in Scotland between 1760 and 1770, and again found no reference to Andrew, Euphemia or Robert in Bowden (which does not surprise me because as far as Scottish baptism and marriage records are concerned, FS/IGI is an index to the original records at Scotland's People). The marriage of Robert Thomson and Agnes Elliot in 1750 is on SP, recorded in both Bowden and Selkirk, which means that the couple lived in different parishes so the banns had to be called in both. None of the indexes records any children of Robert Thomson and Agnes Elliot. There are 14 recorded baptisms of Robert T*m*s*ns in Scotland in 1722, none in Bowden and none with father Samuel.
So the first thing you need to do is clarify exactly what the microfilm you found is. Is it a film of a family bible (reliable) or taken from someone else's tree (unreliable)? Or one of the tiny number of surviving church records (for example episcopalian) that are not on Scotland's People (reliable)?
As for the alleged second marriage to Agnes Elliot in the USA, the most likely explanation for that is that someone has misread 1750 as 1759, and assumed that it must have been in the USA*. The original church records are of the proclamations of Banns, which are consistent with a marriage on 5 February 1750 - viewing the original document may clarify this, although it's quite common for the parish register not to give the date of the actual wedding ceremony. (I think, though I could easily be mistaken, that church rules forbid a second marriage to the same spouse, but things might have been different in the USA, even before independence).
*Yes, I know that the USA as such did not exist until after the wars of independence, but it sticks in my craw to use the term 'America' when I don't actually mean the entire continent all the way from Cape Columbia to Tierra de Fuego, or even the entire continent of North America.
As for the rest of the tree referred to, the politest thing I can say is that it looks pretty unconvincing to me.
The supposed grandfather and great-grandfather look to have been well established in the USA long before the marriage of Robert to Agnes Elliot. I am willing to bet that the earlier generations of it refer to an entirely different family. Norfolk is not by any stretch of the most fevered imagination in Greater London, and George Thomasson (1592-1666, though his gravestone says he was born in 1602) could not possibly have married Katherine Hutton (1612-1646) in 1657, when she had already been dead for 13 years. And how could Samuel Thompson (1691-1753) have married in Virginia in 1717 if he 'arrived' in Virginia 32 years later, in 1749? And if he died in 1753, the same year as his third wife, how did he manage to have two more children with his fourth wife who were, moreover, born in a different state? And why does the tree say that Samuel was born in Blair Manor, Ayrshire when his parents were clearly in the USA long before he was born, and the information linked to his father's grave listing says Samuel was born in King and Queen County, Virginia? (Where in Ayrshire is "Blair Manor"? Does it even exist? Most of the references to it online come from various versions of this same tree, and some suggest it is in the parish of Dalry. There is a Blair **House** in the parish of Dalry, but it is the seat of a *Blair* family not a T*m*s*n* family. )
With so many silly, obvious and easily corrected errors in it, the tree referred to is obviously not just unconvincing, but very unreliable. In fact I'll stop being pointlessly polite and say that as far as the forebears of your Andrew Thompson are concerned, I think it's a load of rubbish and I would strongly recommend ignoring it altogether.