Learn English – What’s the difference between good in math and good at math

Sina is correct that "for which" refers to "all piecewise smooth paths", answering your first question. Specifically, "which" refers to the piecewise smooth paths.

For your second question, "for which" does not mean "because of which". The latter part of the sentence is a description (the generalized definitions... are thus seen to...), and "for which" connects that description to the noun that is described (all piecewise smooth paths, as above). I wouldn't say that "for which" is redundant here because without it the rest sentence as written would be difficult to interpret. But the sentence can be re-written, as in your transformation, in ways that do not require the "for which" phrase at all to connect sentence elements together.

"Expence" is an obsolete spelling of "expense" (like whiche and laste and many, many others) and should not be used. You might see "expence" in archaic documents, but in modern documents it's simply a spelling mistake.

Example from "The Expence of University Education Reduced: In a Letter to A. B. Fellow E.C." (1733)

The Expence requiſite to Education has Limits, and may be known to a certainty within a trifle. The Expence occaſion'd by Idleneſs, or Humour, or Vanity, or Luxury, or Affected Hoſpitality is Infinite, or, at leaſt, hath no other.

Note the long s (ſ) which is pronounced like "s", as well as the capitalization of certain nouns, both of which no longer appear in modern English.