"Através" is a prepisition with a very specific use, in the sense that, it means "through" ONLY when a physical object is going through another. For example:
"The arrow passes through the target" meaning "A flecha passa através do alvo".
Now, because "olhar" is not a physical object, you cannot use "através" to describe it as going through the window. Most Portuguese speakers avoid using "através" simply because it's too specific, and the preposition "por" usually gets the job done in 90% of sentences. An exception would be in the same sentence I used before, where saying:
"A flecha passa pelo alvo" means "The arrow passes by the target"
In this specific case, there is a difference in using "através" instead of "por", but it still is very subtle.
"To see" is "ver", "vir" is "to come".
In Spanish, both "ella vió (p. ver) a través de la ventana" y "ella miró (p. olhou) a través de la ventana" can be used interchangeably. I feel that olhar and ver should be synonymous in portuguese too when used as in "see through" / "look through".
Google: look olhar look, inspect, watch, eye, pore, see
see olhar, ver, contemplar, observar, examinar, considerar