Does peak shaving reduce the stress on power lines
No, don't say "in order to decrease the stress on the transformers and power lines".  Much better to say something like "to lower overall costs".

Most electrical grids have a variety of power sources which are switched on or turned up and down according to the load.  Some power sources take weeks to turn on and off, some take minutes, such as gas, and some are variable, such as solar and wind.  So typically you have a despatch order with the slow things always on and the fast-response thing -- gas -- turned up and down to match load.

I've always heard peak shaving's main purpose as saving you turning the gas on too much because it's expensive.  (Normally calculated as cost of fuel, but increasingly also CO2 or tariffs.)  You might also do it to improve overall reliability, but this is often considered to be simply a cheaper way to achieve a target reliability.

Anything like load-related wear-and-tear will be "just money", and factored into the peak shaving the same as anything else.  From my understanding, almost nothing counts compared to the cost of fuel and capital expenditure on building and decommissioning.

My understanding is load affects cost mostly through different efficiencies at different loads, as opposed to wear, but there must certainly be portions of the grid which do wear out.  There's a chapter on efficiencies and costs in the excellent Copper for Busbars – Guidance for Design and Installation, David Chapman and Prof. Toby Norris, The Copper Alliance Pub. 22, (2014 ed., 103 pp) Download page.