After a bit of research I found that the answers to your question in the original scope could fill a library. And apparently do: or at least a whole museum.
Namely the School Museum in Dresden
Based on your comment, I'll limit the scope to the education system in Saxony during the Weimar Republic. I found just the resource for that:
Die sächsische Schulreform in der Weimarer Republik
or
The Reform of the Education System in Saxony during the Weimar
Republic.
This is the dissertation from 2014 of Andreas Reichelt who studied at the Technical University Dresden.
He quotes a plethora of the actual laws of the period as full sources. It's a 600 page tome and I'm barely scratching the surface. Within these is the
Entwurf eines Volksschulgesetzes, erarbeitet im Ministerium des Kultus
und öffentlichen Unterrichts (vom 12. Januar 1912)
Meaning
The draft of a primary school law, created in the ministry for public
education (*)
Paragraph 36 is
Lehrerbildungsanstalten (educational institutes for teachers)
which states
Zur Ausbildung der Lehrer und Lehrerinnen werden besondere
Bildungsanstalten (Seminare) unterhalten.
meaning
For the education of teachers [using both the male and female forms of
the noun separately special training institutes will be maintained.
This specifically includes men and women, making no distinction of separate paths or, as shown next, prerequisites:
Wer zum Lehramte zugelassen werden will, muß die nachfolgenden
Prüfungen bestanden haben: a) die Schulamtskandidatenprüfung, die
nach erfolgreicher Beendigung des Lehrganges im Seminare von dem
Lehrerkollegium unter Vorsitz eines von der obersten Schulbehörde
bestellten Kommissars abgehalten wird, […]
meaning
Candidates who want to be permitted to the office of teacher, must
pass the following examinations: a) the school board examination for
candidates, which is conducted after the successfull completion of the
course at the seminary in front of the teacher's board led by a
commissary assigned by the highest office in charge of schools
Yeah, I've tried to capture the stilted official German that is even worse than bureaucrat language nowadays.
Note how this speaks of male and female candidates who wish to become teachers and the examinations they must pass which are not segregated by gender
Going by this, there doesn't seem to be any distinction in the requirements to even attend the course. This law came into effect during the Weimar Republic, though it had originally been drafted in 1912.
However, there is also
Gesetz, die Abänderung des Gesetzes über die Gymnasien, Realschulen
und Seminare vom 22. August 1876 betreffend (vom 14. Januar 1913)
or
Law for modification of the law concerning "Gymnasiums, Realschulen and
Seminaries" [this isn't the place to explain the German multi-partite
school system so I'm not even going to try and translate those]
which states on seminaries for (female) teaching candidates that
Aufnahme. (1) Die Aufnahme von Zöglingen erfolgt in der Regel nicht
vor vollendetem 14. Lebensjahre uns setzt eine Vorbildung voraus, wie
solche im Durchschnitt die mittlere Volksschule gewährt.
meaning
Paragraph 69 – Admission (1) The admission of [students] generally
does not occur before the age of 14 and requires as prerequisite an
education which on average is achieved in the "intermediary primary
school".
What constitutes this "intermediary primary school" is not clear, except from context elsewhere which describes that the teaching seminaries last 6 years, so a student would switch over, possibly as a result of some examination results, but wouldn't have any form of "high school diploma" as the expectation was this would be the outcome of becoming a fully trained teacher.
Also
Gesetz über die Ausbildung der Volksschullehrer und
Volksschullehrerinnen (vom 4. April 1923)
meaning
Law on the education of primary school teachers [using again both the
male and female forms of the noun] from 4. April 1923
states
Die Volksschullehrer und -lehrerinnen erhalten ihre
wissenschaftliche Berufsausbildung an der Universität Leipzig
und an der Technischen Hochschule Dresden, ihre
praktisch-pädagogische Ausbildung an mit diesen Hochschulen zu
verbindenden Pädagogischen Instituten.
which means
Male and female teachers receive their scientific professional
education at the university of Leipzig and the technical university
Dresden, their practical-pedagogical training at affiliated
pedagogical institutes.
But makes no mention about admission requirements – except that, with the training applying to both genders, they would most likely have to also match.
In summary, at least in Saxony during the Weimar Republic, the requirements and training for male and female teachers were identical. Interestingly, the laws frequently go back to the later years of the Kaiserreich, where the reforms at least existed in draft form.