1 - as the second reply suggested, you may want to stick close(er) to Berlin. Some smaller cities which aren't too far off are Weimar, Leipiz, and Dresden. For an even smaller town, try Eisenach. There's a medieval (13th century, I believe) castle "Die Wartburg" where Gutenberg translated the bible into German and Bach was born there as well. I've not tried the train connection, but I think it's bout two hours' drive from Berlin. If you want to go far away, there are towns in Bavaria which are gorgeous (Bamberg, Rothenburg, Regensburg, Augusburg, Passau, or anything on the 'Romantic Road') or Baden-Württemberg (Tübingen, Stuttgart, Freiburg i.Br., Baden-Baden, Heidelberg--BW or RLP?), or Reinland-Pfalz (Mainz, Trier, Koblenz, any of the Rhein towns, Boppard, St. Goar, etc.). Or you could go north from Berlin to the coast, to Rostock, Rügen (Island), Heiligenhafen, Lübeck, Greifswald, anywhere up there...
2 - That will depend on the time of year, which day, and which part of Berlin you're in.
Prenzlauer Berg: ,,Markt am Kollwitzplatz" (Food)
Sat / Sun 9 to 4, Thus 12 to 7 is organic
U Senefelderplatz
Trödelmarkt am Tiergarten (Art & Antiques)
Straße des 17. Juni, Tiergarten
Sat / Sun 10 - 5
U Tiergarten
Schöneberg: Winterfeldmarkt
Winterfeldtplatz,Schöneberg
Weds / Sat 8-13
U Nollendorfplatz
Turkish market
Türkischer Markt am Maybachufer. Authentischer Straßenmarkt.
Maybachufer, Neukölln
Thurs - Fri, 12-16.30
U Schönleinstraße
http://www.plazahotel.de/shopping_in_berlin.aspx?lang=en