Pax vobis,
Is the preposition “per” is kind of like an all/multi purpose preposition? Can it be used for the various English prepositions: for, by, from?
Gratia vobis,
per means through in terms of space and time. With time the accusative alone is more commonly used but “per septem dies” for seven days is more emphatic. If you wish to use ‘by’ in the sense of ‘by means of’ a simple ablative will indeed be less rare than a per + acc. which can however happen. ‘by’ as agent will require a simple ablative if an inanimate object or a/ab if a person as in tua mater ab Diana necatur. The idea of for/to is conveyed most often by a simple dative, but when for means on behalf of you should be using pro + abl. Compare ‘vobis est hoc donum’ (this present is for you/is yours) and ‘pro vobis modo officio fungor’ (I am just doing my duty for you all). From is e, ex when you come out of something, a/ab when just from the edge and de from the top i.e. down from, all of course +abl.
There are so many other uses/deviations from such simple rules in Latin.
Gratia tibi