Questions from The Red-Headed Leagues [closed]

A narrower definition of propagation is the one that you supplied, i.e. the sense of growth only in number or count. 'Spread' specifies the additional meaning of being distributed more widely across some physical space or geographical area. In some cases this would be redundant and the 'spread' meaning is already implied by 'propagate', but in this context of red-headed people you could have a family of impoverished people having many children but all living within the same house, who would be propagating but not necessarily spreading. By specifying 'spread' it conveys the intent for those individuals not only to increase in number, but to prosper or succeed or somehow have greater influence throughout society.

Yes. For an example of the word being used in this context Singapore has a "Maintenance of Parents Act", a law related to families making sure that their elderly relatives have sufficient care in terms of meals / shelter etc, not that the parents need to have regular repairs/servicing like a piece of machinery (the more typical use of the word "maintenance").

This usage of "should" is equivalent to "It is exceedingly unfortunate that you happen to be a bachelor" or "It is exceedingly unfortunate that you are a bachelor". The creator of the fund supposedly wants the red-headed 'gene' to flourish ('propagate') which obviously won't happen if the man is single and not having children. This other question is more focused on this usage of "should" https://ell.stackexchange.com/questions/90189/is-there-a-difference-in-meaning-among-should-happens-to-or-should-happen