In general, both yes and no.

What I mean by this is that it is extremely hard to reconstruct the genealogy of people of African descent beyond a few generations of slavery, and ever harder to trace the origins of those who were originally enslaved - at least by traditional archival methods. The most obvious reason for this is that the best available surviving records (ships logs, etc.) very rarely go into the particulars of listing the name and place of origin for slaves they transport to North America. Few records from the enslaved exist, and even fewer make mention of a specific location they were from.

But, with the availability of DNA tests to show genetic heritage, it has become much easier to determine with some certainty where ancestors came from.

To go back to your first question, on what countries existed, it is again complex. The Atlantic Slave Trade began in 1501 and lasted through to the mid-19th century (it was banned/suppressed by International Treaty in 1807/8, though smuggling continued after). In that time the boundaries of states in Africa quite often changed, in many cases due to warfare that the slave trade encouraged and thrived off of. You can get an idea of these changes by viewing older 18th-19th century maps.

Now what we do know is that a vast majority of enslaved Africans originated from the west coast of Africa, an area roughly spanning from modern day Senegal southwards to Angola. From these locations, the largest majority (apx 44.2% between 1519 and 1867 per Eltis et al) of slave voyages departed from West Central Africa - an area comprised of modern-day Gabon, Republic of the Congo, Democratic Republic of the Congo, and Angola.

Sources:

For statistics see "The Volume and Structure of the Transatlantic Slave Trade" Eltis, David et al.

You can also see the National Park Service Ethnography Report on Africa which provides maps and details.