First question re the languages the barbarians spoke: Depends on the precise barbarians.

• Huns spoke early Hunnish and Gothic languages plus various other tribal languages. Hunnic is not well recorded enough to know what kind of language it was, although scholars favor Turkic.
• The Goths spoke Gothic, which was a East Germanic language with various dialects. With early documents from the 6th century, it's the earliest Germanic language known, but has no modern descendants. The Vandals are also believed to have spoken an East Germanic language, but no trace remains.
• By the 5th century A.D it seems West Germanic was well distinguished from East Germanic. Proto-Germanic is thought to have become distinct around 1,000 years before at least, around 5th century BCE.
• French itself came in part from Old Frankish, a Western Germanic language mixed/replaced by Latin when it was under Roman rule (Roman Gaul), which to my eye explains some quirkiness about modern French relative to Latin. Old Frankish also fathered Dutch. Old Frankish was spoken between 4-8th century AD.

As for the origins of Proto-Germanic, you're asking for a thumbnail of information that is widely available, but the details are still disputed as research progresses.

Proto-Indo-European (PIE as commonly abbreviated) is the root of most European language families, other than Basque which is unrelated to any other language living, or Finnish which is an Uralic language. It's also the root of Indic languages like Sanskrit and its modern descendants, Hindi and Hindustani, Urdu, Punjabi.

PIE was reconstructed from comparing living Indo-European languages. Through comparative analysis and phonetic analysis, we have established the original language knew horses, wheel, and specific trees, suggesting a northerly central Asia location, aka the Pontic-Caspian steppe, with the original peoples probably being at least partly steppe herders.

There were other languages before PIE-root languages became established in Europe, such as Etruscan, with remnants only in place names and a few plant names and some loan words.

Archaeology is tentatively linking the origin of PIE to the spread of the Yamnaya culture (also called Pit Grave culture, or the Ochre grave culture) into Europe. This culture is identified by characteristic artifacts. This is a hot field of research and dispute, though.
Disease may have played a huge role in the spread of PIE and its daughter languages. https://www.sciencemag.org/news/2018/12/did-new-form-plague-destroy-europe-s-stone-age-societies

Because PIE apparently spread fairly quickly into Europe and mixed with the original Europeans without central governance, the separate language families may have developed pretty quickly, in mere generations, just as languages/dialects diverged quite quickly in Europe as the Roman Empire declined and fell.
The relationships are normally visualized in a language family tree, so you can see the splits. Minna Sundberg did a lovely illustration that has been republished many times. I'm linking to the Guardian Article featuring it. https://www.theguardian.com/education/gallery/2015/jan/23/a-language-family-tree-in-pictures

As for the date of PIE, it is hypothetically dated back to 4500 to 2500 BCE, but estimates vary. Could be earlier. Not likely to be much later. The earliest form of ancient Greek is dated to 1500 BCE re linear B inscriptions. It does have intermediate characteristics between PIE and later Greek, but already had distinct sound changes that are indisputably Greek.

1000 BCE for Proto-Germanic may be around right. Maybe a few centuries either way.