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1. Landlords decide individually what they do or don’t require before renting out a place to you. Most would want to see a proof of income, to be sure you can actually pay the rent.
Landlords might be more flexible with this in the Eastern German cities, where there are many unrentable, empty flats (but far less jobs). Or you alleviate their fears by offering to pay a year’s rent in advance.
2. Job chances and search strategies depend on your industry, educational level, etc., of which you mention nothing - so we cannot advice further.
Of course language skills are the most important factor for finding any job. The address, however, plays only a minor role.
3. As an EU citizen, you can just move and register your new residence (at the local town hall) within a week.
You must then join a German health insurance within three months (they will retroactively charge you from the day you arrived, so no point procrastinating here). If you join the public scheme, this will initially cost you between €350 and €750 (depending on your world income, which they will estimate). After you start work, half of this is paid by your employer. If you are single, young and healthy, a private insurer might be cheaper - get expert advice once you’re here, as this is a complicated topic with wide implications.
No formalities are required after that (at least until you found a job).
Don’t forget to bring all your documents (from birth certificate to last employer’s references) as originals - you’ll need them sooner or later!