Not in all instances, no, and in particular, not in this one. In the majority of cases, "on foot" is idiomatic and "by foot" isn't. You will occasionally see "by foot", but it's mainly limited to cases where someone is on, like, an expedition. "They trekked all the way across the Amazon by foot", "He went from city to city by foot, interviewing people"
Wow 9 months, and nothing happend? GEEZ I wonder why! Somebody should maybe go quickly "ON FOOT" (by cutting off someone's foot) to somebody in charge her! Oh wait, nobody cares about the Russian section, or maybe people do and they just wanna piss off people on purpose by just ignoring obvious mistakes.
Why "seldom" is wrong in the sentence "she seldom walks to work"? As far as I know, seldom and rarely are synonyms.
пешком seems to be an odd word.
Пешка is a feminine word for "pawn" (a chess piece), which declines regularly. The instrumental for пешкa is пешкой not пешком, which is either masculine or neuter, so the word "пешком" doesn't seem to be based on anything, though related to the chess piece.