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How hard is it to avoid "adult" stores and themes? Does it depend a lot on the neighborhood? This is my first time visiting Japan so I don't know how accurate this is, but I have heard that "adult" stores are mixed in with other everyday stores speaking from an American perspective here in sort of an obvious way.
Similarly, that it is not unusual to run across what to an American sensibility would be strongly inappropriate material in manga or similar on pretty open display in stores. I'm not here to litigate whether that's good or bad, right or wrong. I'm the visitor, so however the locals roll, they roll. That said, I'm traveling with a tween boy and while I wouldn't call ourselves puritanical, I think we have a fairly old school approach to these things.
He's interested in and curious about manga and similar things, and we'll be walking around different neighborhoods to visit. We like to explore and wander. I definitely like to wander around Shinjuku see the big neon lights and all but would prefer to have long conversations with my son about sexual mores, etc. You would probably want to avoid the Kabukicho section of Shinjuku , but that's just a small area of Shinjuku.
You could use google maps street view function to take a tour of the area so you'll be prepared. I have stayed many times in the Ikebukuro neighborhood, which has many adult entertainment establishments and short-stay "love hotels. There are some signs on storefronts indicating "nobody under 21," the love hotels have signs showing a variety of rates for hourly or longer stays, and you might see a few signs with pictures of women, but those are the only hints of anything "adult.
If your kids ask questions about the girlie bars, you can tell them that they are places where businessmen go after hour workdays to have drinks with women who listen to their dull conversation and flatter them, and that would not be far from the truth. Its not super hard to avoid this stuff. In akihabara some of the stores the upper floors are to be avoided because of the adult section. You will want to avoid "entertainment districts" displaying photos of women and hourly "system" prices.