If I purchase additional insurance, can I drive the rental into Mexico?

Last time I was there there was signs and notices about not driving in Mexico. And warnings besides rental car companies about how risky Tijuana was. Might want to also check US state department web site advisories. Call Avid and ask what the current policy status is.

Yes with some stipulations

Yes, but you have to get it through an Avis rental facility. Also not all all facilities offer the additional coverage. I had to go to the San Diego airport facility to purchase a policy just for Mexico.