I can provide only mostly speculation for the first two, but,
it seems the reading しめ for 七五三 comes from the word しめなわ (しめ縄), a type of rope described as such:
The shimenawa is always made of straw twisted to the left, the pure or fortunate side, with pendant straws at regular intervals but of differing numbers in the order three, five, seven, along the whole length of the strand.
To be honest, I'm not quite sure what that description means, but the bottom line is that the numbers three 三, five 五, and seven 七 are characteristic numbers of しめなわ and so the word is sometimes written 七五三縄, in which the 七五三 would be read しめ. Is that clear?
For 孫右衛門, I would imagine that the word was once pronounced まご
うえもん, or perhaps even まご
うえ
いもん, and that some of the vowels disappeared over time, but it retained its kanji spelling.
For your last question, as you may already know, の is an attributive particle, which in a name would serve a purpose similar to "de," "mac," "di," or "von" in various western names, or "upon" in several English place names. And being simply a function word, it is omitted in certain place names when they are written in kanji, though the pronunciation retains the の.
I should also note, that for a period of time, Japanese was written entirely without kana (indeed, there was a time when kana didn't exist), and readers would have to fill in particles and verb endings in their heads. Even today, there are times (and names are a prime example), when inflections and particles are left unwritten for the sake of writing something with only kanji.