I was in Berlin a few months ago. I did not get an overt anti-american vibe. As for how to dress, can't say. We got picked out of a crowd as a tourist by every bootleged purse salesman or shell-game huckster around. People saw us and defaulted immediately to English. So, i have no good advice on that!
I'd avoid Berlin though, I found Muchen, Rothenburg ob der tauber, and Nurmburg to be far more interesing. I suppose it depends on what you're hoping to see while there.
I recommend trying a half-meter-wurst, and curryworst. Don't try too hard to eat at the hofbrau house in Munich... There's a restaurant kitty-corner to it that was much better.
Ditto on skipping the Hofbrau Haus. The kitty-corner restaurant is called "Der Alte Fritz" and it is well-regarded for traditional German food.
That said, I really like Berlin! It is very much of mix of old and new, which a strong shot of Cold War stirred in. There are a lot of good mid-priced hotels around Alexanderplazt, in the former East Berlin which is a good central, place to be with lots of tourist activities/things to see in easy walking distance. All of the museums on Museum Island are excellent. The DDR Museum (same area) is also surprisingly interesting. The Technology Museum (not in the same area) is huge, and will take more than one day to see, so plan accordingly if that's your thing. You could spend an entire week in Berlin just seeing museums, there are lots more and very good.
The Raddison Blu has a giant five-story salt-water fish tank in the lobby that your kids might enjoy. Just walk in like a paying guest and have a seat. The Park Raddison has top level observation deck (costs a couple Euros) that has spectacular views of Alexanderplaz. An excellent walking tour (available guided, or you can just do it yourself) is to start at Alexanderplatz and simply walk to Museum Island (you can hit the Raddison Blu on the way) and then down Unter Den Linden to the Brandenburg Gate and the Memorial to the Murdered Jews of Europe (must see). There are interesting sights the whole way. If you like, you can continue onto the Reichtstag, but a reservation is required to go to the top (it is cool, but safely skipable if you don't have time). Checkpoint Charlie is definitely skipable. There are several sections of the Berlin Wall still standing, which a must see, IMO. One of the most famous sections is the East Side Gallery where artists have painted murals on the wall. Potzdammer Platz is definitely skipable. It just a modern section of the city that has the same stores as every other big city. If you want to just wander around the neighborhoods, the Mitte and Prenzlauer Berg still have a lot of the old architecture and lots of interesting shops and restuarants.
It is possible to eat cheaply in Berlin. One way is get a currywurst from street vendors, a hot dog with curry ketchup, definitely a Berlin thing. Another is the doner kebab, which is a Turkish kebab done Berlin-style. Every place does it slightly differently, but you can't go wrong anywhere and there are tons of doner kebab joints. When I travel, I try to eat the food of that country as much as possible. I feel like that's part of the experience of traveling. That said, it is harder than you might think to find traditional German food, but you can find any other kind of food you want. A good old-school traditional restaurant is Der Alte Fritz mentioned above, which is right off Alexanderplaz. Another is Max and Mortiz which is a traditional, yet laid back locals place in the Kreuzberg neighborhood. And Die Schwarzwaldstuben in the Mitte neighborhood, which is lively, modern take on traditional cuisine sort of place. Again, more of a locals joint.
That's just scratching the surface. Let me know if you have more questions. I'll do Prague in another post.