No, at any point in history there are only ever one or two stargates on Earth.
From pre-history there is one gate in Antartica, emplaced by the Ancients, which we'll call Gate 1. At some point they ascended and the gate fell out of use. The Goa'uld arrived millions of years later and emplaced their own gate which we'll call Gate 2.
The SGC discovered Gate 2 ...
I have read that there is a Tok'ra harcesis.
I have read about a woman called Ser'name, she is the daughter of Kadon (a man who host of Selmak previous to Saroosh) and Pria (the host of Ja'nok); I have read that Ser'name Harcesis gene has been deactivated, so Ser'name cannot access to her Harcesis memory.
The Ancients repopulated the galaxy after the plague that wiped most of them out using the Dakara superweapon. Humanity in the Stargate franchise is the recreation of the Ancient civilization, which is the reason why Humanity is the fifth race.