"We pay for the box of cookies."
Translation:Paghiamo la scatola di biscotti.
You can, but it has a different meaning: "paghiamo per la scatola di biscotti" indicates either causation (e.g. our kid cracked open the box of cookies, so we pay for it) or partition (we bought several things, but we pay for the box of cookies, others will pay for the rest).
The root of the verb is "pag-", with a "hard" G; but adding to it the ending -iamo it would change the sound to a "soft" G (pronounced as in "page"), so the H is added to keep the letters separate. The same rule applies to other mutable parts of speech too, e.g. "la banca" (the bank) / "le banche", "il lago" (the lake) / "i laghi", "poco" (few) / "pochi". Unfortunately some roots are "weak" and actually change sound, e.g. "lo psicologo" (the psychologist) / "gli psicologi", but you can consider them as exceptions and memorize them when you meet them.
Yes, you could say that. By the way, a similar phenomenon sometimes happens in reverse, in nouns and adjectives ending in -cia or -gia in the singular: if the "i" isn't part of the root, but only there for the "ch" or "j" sound, it gets removed in the plural, e.g. "l'arancia" (the orange) / "le arance", "la coscia" (the thigh) / "le cosce".