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The rental market in Germany is characterized by long termination notice times (minimum 3 months, by law) and especially in the bigger cities with tighter market conditions (Berlin is among them) flats are advertised an quickly taken up soon after termination notice is given.
Thus you cannot reasonably expect to find something ready for move-in soon - especially if you have requirements (dogs allowed) that rule out a major part of the market.
I thus strongly recommend you arrange temporary accommodation (hotel, hostel or serviced apartment) for the first few months until you find a permanent place.

Basic house pets like cats and dogs - unless somehow unusually large and dangerous cannot generally be legally excluded from rental property in Germany.  A landlord could theoretically not like that you have a dog and thus not take you - but they cannot legally just disallow dogs. I am posting a link below to a site (in German) that gives details. There are lots of such links if you know what to look for. Yet this law is not known to all Germans (from his response I suspect this is the case with Beppi – who otherwise gives good advice) and some landlords might try to disallow pets not knowing that. Exceptions to the law are seen on a case by case basis. For example, if a landlord lives in the same house and is allergic to dogs - or if an individual dog is known to have behavioral problems like attacking people or barking all of the time - this could give a legal ground to not allow it.

It is good to know the law as a reference point but the reality is that if a landlord is dead set against animals then you have little chance to win in front of a court. It takes years, lots of money and they can just claim other reasons as justification. But because of the law and the underlying attitudes of society, a dog should not be a major problem in getting a place to live. Maybe a cute dog will be a plus with some landlords! The problem is if the landlord feels secure in renting to a foreigner. This will depend a lot on your economic and job situation, appearance and how you communicate.

In addition to what I wrote above, maybe it is best not to mention the dog. They cannot later say it is not allowed or that you should have mentioned it. Even if there would be a clause in a rental agreement saying no pets - this would be invalid according to German law.