I cannot help feeling grateful for my parents' loving kindness in having let me spend my boyhood in the country. If I had gone with them to live in Tokyo at the age of eight, I think I should be quite a different sort of person from what I am today.
Are there any errors in these sentences?
It's fine - perhaps too carefully correct! I think most British native speakers would say '...in letting me spend...', and the 'I should' suggests extreme (prescriptive) correctness. Most people, in my experience, would say 'I would'; "I should" is certainly not wrong though.
On a personal note, I would say 'quite different from the sort of person I am today'.