Elly, there is no Gera in the state of Saxony (in German: Sachsen; capital: Dresden). There are, or better said, were two towns called Gera in today's state of Thuringia (in German: ThÃ¼ringen; capital: Erfurt). The name of one of them changed a bit 81 years ago.
The larger Gera is in eastern Thuringia and was the capital of what was until 1918 the very small Thuringian Principality of Reuss Younger Line (in German: Reuss jÃ¼ngere Linie).
The smaller Gera was in western Thuringia in what was until 1918 the small Thuringian Grand Duchy of Saxe-Weimar-Eisenach (in German: Sachsen-Weimar-Eisenach), just outside the city of Ilmenau. (Like all of the small Thuringian states, Saxe-Weimar-Eisenach was made up of enclaves not contiguous with one another, and one of those enclaves was the city of Ilmenau and environs.)
The Gera to which you have already been referred by somebody here on this board is the WRONG Gera! So forget that, or your research will be thrown off entirely.
If your source indicates that your grandmother's birthplace was "Gera, Sachsen", then she was born in the OTHER Gera; that is, the Gera in western Thuringia near the city of Ilmenau in what was until 1918 the small Grand Duchy of Saxe-Weimar-Eisenach, the "Sachsen" coming from the state's German name, "Sachsen-Weimar-Eisenach".
In 1920, two years following the end of World War I and the abolition of the German monarchies, the seven small Thuringian states, which had been such a confusing muddle, merged into a new state called Thuringia (in German: ThÃ¼ringen), with the city of Weimar as its capital. In 1944, the Prussian enclave of Erfurt was transferred from the state of Prussia to the state of Thuringia. The city of Erfurt is Thuringia's capital today.
In 1923, the Gera where your grandmother was born merged with the nearby town of Arlesberg. The combined town was given the name Geraberg. So today's Geraberg is the town you're actually interested in.
As to records, you can contact the town administration (in German: Gemeindeverwaltung) to see if they have birth, marriage and death records on file there. There's a possibility that those may be on file in Ilmenau, but contact Geraberg first:
Going on the assumption that your grandmother was Lutheran, you can also contact the Evangelical Lutheran church in Geraberg and enquire about church records:
You also asked about the Mormon International Genealogical Index (IGI). All information in the IGI is contributed by people just like you and me, which is why it's often unreliable. A person has some information and asks the Family History Library out in Salt Lake City to add that information to the IGI. The Family History Library then does that, no questions asked. The Family History Library does not verify any of the submitted information. The fact that information has been submitted to the IGI about your grandmother's family indicates that there's someone else somewhere, probably some kind of a relative of yours, who's also researching her family. (The information was not submitted by someone who works for the Mormon Church, but rather, by a member of the Mormon Church.)
Hope this is of some help to you.