I'll break this down into individual answers.
Sorry my post is confusing. I'll explain.
CyanogenMod is a custom ROM for android devices. (Please note: do not take this as an "oh, this is awesome, go install it!" scenario. I don't need people breaking their stuff.)
I do have a nexus 7 2013 Wi-Fi, and bandai's app detects that I am running an operating system that is not default android with extra privileges enabled. For this reason, it refuses to open, and gives me an error message.
If you are running regular android out of the box, you *WILL NOT* have this problem. Strangely, it is the only app I have installed that denies me access. Bandai are being childish about this.
Thankfully, Mr Blinky's app works on my device, because he took that "check" out of the English app.
About compatibility, this will be confusing again, but I will explain it the best I can.
There are a few different types of NFC. Most phones and tablets have the regular NFC standard built into them. The 4U uses a special NFC standard called FeliCa, which is different than regular NFC standards, and very popular in Japan. (Reference)
https://en.m.wikipedia.org/wiki/FeliCa
There are very few devices currently with FeliCa as the NFC chip. Most of them use standard NFC. That's why the compatibility list is so small. But lucky for us, the Nexus 7 (2013, not 2012) uses it, and happens to handle the app very well. That's why the phones are expensive. They use a different NFC standard.