Is Earth the home planet of the Culture?

No, Earth is not the (or an) originating planet of the Culture.

Consider Phlebas is chronologically the first Culture story, and it takes place in 1331 CE according to its appendix "A Short History of the Idiran War" (page 467) and the fact that the Idiran war is in its fifth year (page 19).

So the Culture exists concurrently with human society on Earth. The Culture visits Earth covertly in the 1970s (The State of the Art) and is assumed to formally make first contact with us around the 2100s (the aforementioned "A Short History of the Idiran War" formed part of an independent, non-commissioned but Contact-approved Earth Extro-Information Pack, as mentioned in said appendix).

Iain M. Banks also wrote "A Few Notes on the Culture" which explains some things about its nature, including the fact that it was founded about 9000 years before our present.

The Culture is a group-civilisation formed from seven or eight humanoid species, space-living elements of which established a loose federation approximately nine thousand years ago. The ships and habitats which formed the original alliance required each others' support to pursue and maintain their independence from the political power structures - principally those of mature nation-states and autonomous commercial concerns - they had evolved from.

It's known that the Earth is not the Culture's original planet as the Culture covertly visits Earth in The State of The Art in our 1970s.

By the spring of the year 1977 AD, the General Contact Unit Arbitrary had been stationed above the planet Earth for the best part of six months. The ship, of the Escarpment class, middle series, had arrived during the previous November after clipping the edge of the planet's expanding electro-magnetic emission shell while on what it claimed was a random search.

The Culture doesn't have a single home planet. It was originally comprised of several dozens worlds that formed a loose, then close alliance.

We learn the name of at least one of these worlds in Excession.

For what it was worth after so much time, a drone team had visited the preserved battle site itself and conducted their own deep-scan of the ground. The fact that Xlephier Prime was one of the twenty or so planets that could fairly claim to have been one of the home worlds of the Culture - not that it really admitted to having such things - made the task easier.

In The State of the Art, we learn that Earth exists in the Culture-verse, but wasn't contacted until the 1970s:

At five minutes and three seconds past three AM, GMT, on the morning of January the second, 1978, the General Contact Unit Arbitrary broke orbit above the planet Earth. It left behind an octet of Main Observation Satellites - six of them in near-GS orbits - a scattering of drones and minor missiles, and a small plantation of young oaks on a bluff near Elk Creek, California.

In Excession, (taking the date from The State of the Art) we learn that the Culture has been "in play" within the Milky Way since at least 2000BC.

Dreve would be an ideal system to do this in; it had been a Culture system for four thousand years, comprising nine more or less wilderness worlds and three Orbitals - hoops, giant bracelets of living-space only a few thousand kilometres across but ten million kilometres in diameter.

And from The Hydrogen Sonata that it's existed as an entity for over ten thousand years.

The Gzilt were a sort of cousin species/civilisation to the Culture. Nearly founders, though not quite, they had been influential in the setting up and design of the Culture almost ten thousand years earlier, when a disparate group of humanoid species at roughly the same stage of technological and societal development had been thinking about banding together.