A general word of advice: German authorities are bureaucratic. You cannot expect to go there with the documents you think you need and get married immediately. Even if you do by chance have the right documents (see below) you will make an appointment for an appointment to be married.
Should we wait until she receive the extension of the visa? Should she apply for another visa?
Go and ask. If her visa is valid right now and at the date of the actual marriage I don't see a problem, but you don't know when that date is... so please don't guess, go and ask.
What is considered a valid proof of residence?
Normally, a rental agreement. You live somewhere, you have a contract. Bring a copy and you should be good.
Ehefähigkeitszeugnis is issued by the german administration, or should we apply for it in our respective embassies?
Now the real fun starts. There are two types of countries: countries that trust their citizens and countries that do not. Germany is not very trusting. While in other countries you swear that you are legally allowed to marry and that's it, in Germany you will need paperwork. You need papers, certifying that you are legally able to marry (i.e. not married, if previous marriage existed then divorced etc). The fun starts when you are from a country that is trusting their citizens because they will simply not have such a certificate. In that case, Germany will need to see other papers they accept instead and obviously that will vary by country.
So do yourself a favor: go get an appointment with Standesamt as soon as possible, talk to a real person and get a definite list of papers you need. There is no way you can navigate this on your own. People at the Amt are friendly and helpful if you come prepared and expect to leave with a checklist to do some work. They can be very unhelpful when you just go in with papers and want to get married. Because you will not have the right papers.
So go and ask. Work with them. Let them present you with a list of what you need and then go get it. Be prepared that this may take several weeks and several hundred € (because you will need a certified translator to translate all of it) just to get to the point where you are allowed to make an appointment to marry.