a gene is passed on from generations and a chromosome is just found in certain cells

nucleus → chromosome → gene

Smallest to largest:

Gene (a place on a chromosome); chromosome (there are 46 in human cells); and DNA (because it accounts for all the genetic material in a cell).

no, the gene is not actually the chromosome but a part of chormosome...

There should be minor difference in ambidextrous of redheads to other hair colour because the red hair gene is on chromosome 16 while the dominance of hand is the work of gene in possibly many chromosome. For more specific, chromosome 2 is in favour of developing left-handedness but not always be true. Even if there is a supportive factor of ambidextrous in chromosome 16, the other gene might simply overrule the supported gene in chromosome 16.