Did God invent Hebrew?
No, humans invented the Hebrew language. Hebrew evolved out of Old Canaanite, and has been a distinct language probably for at least 12,000 years.
Our tradition states that Hebrew was the language with which God created the world (Rashi commentary, Genesis 2:23, quoting the midrash). Since it was considered a holy language and was used for prayer and the teaching of religious tradition, it was not spoken in mundane contexts and wasn't taught to just anyone. It was handed down from individual teachers to disciples as part of the original tradition; and the same goes for the art of writing. Thus, certain Hebrew Psalms (92 and 139) and teachings are attributed to Adam, the first man. The wider public, most of whom descended relatively quickly into idolatry and sin, were not given access to the treasures of the original tradition, since by their actions they implicitly repudiated it.
After the Flood, the Hebrew language had a brief period in which it was generally known, thanks to Noah (see Rashi commentary on Genesis 11:1). This is why many hundreds of Hebrew words have cognates in languages as diverse as German and Japanese. The alphabet, which secular scholars trace back to the Phoenicians, is according to our tradition actually one step older than that: it is from the Hebrew aleph-bet, which those of the Phoenicians and Greeks closely mimic.
After the Flood also, the knowledge of Hebrew eventually declined (see Genesis ch.11) and was preserved only among the Western Semites, the ancestors and cousins of Abraham. Eber, from whom our word "Hebrew" (Ivrit) is named, was a Semitic descendant of Noah and ancestor of Abraham. He was one of the major transmitters of the original traditions. He is credited with having broadened the Hebrew language, and some Hebrew grammatical constructs are attributed to him by certain Jewish researchers.
As time passes, languages grow and adapt. Thus today we can identify words and types of usage that go all the way back (and these are the ones that are most likely to have widespread cognates). And then there are Late Biblical Hebrew; the Hebrew of the Mishna; Medieval Hebrew, and so on. All of these have a broad overlap, but each has introduced its added vocabulary words and usages. Today, Torah-Hebrew includes some words that were borrowed from the Persian, some words taken from ancient Greece, Aramaic words, etc.