I am interested in this family (as I am directly related) and feel there is enough evidence to indicate the family lived in or near the Newhills area.
Where was Peter born, and where did he die? Where did you find his birth and death dates? What is the evidence to suggest that he lived in Newhills?
I believe James Smith and Jean Mitchel are Peter Smith's parents
What evidence do you have to make you think that his parents are James Smith and Jean Mitchel?
and Alexr Smith and Bessie Smith (nee Chalmer) are grandparents.
Again, what is your evidence for this?
Bessie Smith appears on many christening records is this Bessie Smith or Bessie Chalmer?
If she was baptised as Smith she is not Bessie Chalmers. She would be recorded as Bessie Chalmers in her children's baptism records.
George Smith is in Newhills at Rivehill in the 1696 Pollable People, is he likely related?
No idea. Where did Alexander Smith and Bessie Chalmers live?
What is Rivehill's present name?
Rivehill.
See
https://maps.nls.uk/geo/explore/#zoom=16.0&lat=57.17979&lon=-2.27142&layers=5&b=1&marker=57.17603,-2.27453 and
http://www.geograph.org.uk/gridref/NJ8309How should I go about verifying all this?
Look for original documents at Scotland's People, Scotland's Places and other primary sources.
For the parents I've found there are James Smith and Jean Mitchel in Madderty, Perth and Patrick, Peter, and Hary Smith are christened there.
The recorded children of this couple are
William, 1758
Hary, 1761
Patrick, 1764
Are these from a different family or the same family?
Without dates it is impossible even to speculate, especially with names as common as Smith and Mitchell. Madderty is quite a long way from Newhills in 18th century terms. You would need to find some pretty solid evidence to show that they were born in Newhills and Strichen, married in Strichen, had their family in Madderty and then the family turn up in Aberdeenshire. Far more likely that the Madderty couple is a different couple with the same names.
There is a marriage in 14 Nov 1749 in Strichen but there is a marriages of James Smith to Margaret Gray in St. Nicholas Aberdeen. Is this a different James Smith?
If he was married to Margaret Gray and your Peter's mother was Jean Mitchel, then he is unlikely to be the same James Smith, unless he was married twice.
The marriage of James Smith and Margaret Gray is recorded in both Aberdeen and Old Machar. This is after the marriage of James S and Jean Mitchel, and before the births of the Madderty children. Therefore (a) if the couple in Madderty is the couple married in Strichen, the husband of Margaret Gray cannot possibly be the same James Smith or (b) in the unlikely event that it is the same James Smith, then your Peter cannot have been born in 1764.
However the Aberdeen couple, James S and Margaret Gray, had two recorded children
Donaldson, 1757
William, 1760
As the eldest recorded child of James S and Jean Mitchel in Madderty was baptised in 1758, these families overlap, therefore it is not the same James Smith.
Not really surprising as Smith is the commonest surname in Scotland and James is the second commonest given name after John. There are 134,151 records of James Smith, not including spelling variations, in the Scotland's People web site.
Smith is a common name but Peter as a first name less common and his christening has mother Jean Mitchel.
Are you assuming that the couple in Madderty are the parents of your Peter? It's never a good idea to make assumptions.
Peter's son Peter was a blacksmith and had a small croft.
Where? And when and where was Peter junior born? And who was his mother?
What is the Associated Church?
One of umpteen splinter groups and denominations that split off from the Church of Scotland. Where have you come across the Associated Church?
For the grandparents Alex Smith married Bessie Chalmer 24 Jun 1709 in Newhills.
Alex S and Bessie C had three recorded children, all born in Newhills
James 1709
Jean, 1718
Alexander, 1718
Would the parents get married in Strichen because Jean Mitchel is from that area? or are they an unrelated couple?
It was customary to marry in the bride's home parish.
I did find clues that suggested Peter Smith wife Mary Philip (1770-1850) father's name was William Philip but have lost the source and it represents a dead end.
Where was Mary born and where did she die?
I see that Peter Smith and Mary Philp had seven recorded children
George, 1795, Kinellar
Peter, 1797, Kinellar
Margaret, 1803, Skene
July, 1805, Skene
James, 1807, Skene
John, 1810, Kintore
Mary, 1812, New Machar
If Mary's father had been William, you'd expect them to name a son William. Though there is a gap between Peter and Margaret that could have contained one or two more children.