Non EU Spouse on student visa switching to EEA family permit

[kirsseb]

I am a German citizen and my girlfriend is Chinese. We are both currently doing a Master's degree in the UK and she is here on a Tier 4 student visa. We both came to the UK in August 2016 and are graduating in September 2017. We have been in a relationship for 9 months now.
Due to a job offer I am considering staying in the UK and working in from September 2017 onwards.
My girlfriend and I are sure that we want to get married and live together. Here student visa is still valid until January 30th 2018.
Right now we are considering whether there would be any way for her to stay here in the UK with me
Since I am currently studying and working from September onwards I am exercising treaty rights as a EU citizen. I would also be earning more than the 18.600 GBP.
So I have a couple of questions in that regard:

If we move in together after graduation in September 2017 and get married soon after (Oktober, November) would it be possible for her to apply for an EEA family permit from within the UK and get it before her visa expires?

How long does it take to actually get married in the UK? From what I have read the process can take quite a lot of time if a non-EU citizen is involved since the Home Office might require an interview and may extend the time until we can have the actual ceremony.

Is this a realistic path? If not what would be the best alternative especially with Brexit approaching?
Another option we considered was for her to move back to China, getting married there and applying for a family permit from China. Any experience for how long would it take for her to be able to come back to the UK in this case? Apparently, the initial 6 month permit is processed within 3 weeks.

Thanks in advance!