The Kobo Android app should sync books that you purchase using the account it is signed in as. It usually happens the first time I start the app after buying a new book. On the rare times it doesn't, or I already have the app opened, I do a sync from the top right menu to get them.
Where are you looking? The books should be in "My Books" under the "ebooks" tab. This can be filtered to only show books in certain states. If you remove the filter and sort by "Recent activity", it should be at the top of the list.
Once you find it in the lists, you will need to download it to the app. Tapping the books cover should do that.
If you are looking in the file system, I don't know where to look. I don't know it they are kept in a normally accessible location. And if you can find it, you may need to deal with DRM.
If the book has downloaded to the Touch from Kobo it should be in your account. You can check that on the Kobo store by going to "My Books" there. From here, most books can be downloaded directly to whatever device you are using. In the case of an Android device, you will be asked how to open the book. You do not need to use the Kobo app for this. And, if it has DRM, it will be Adobe DRM and you need to use an app that can handle that.
When you look at the Touch, books downloaded from Kobo are stored in the directory ".kobo/kepub". The name is not meaningful and you would need to work out which it is if you have more than one book. And if it has DRM, you will need to open it through the devices UI. Copying this to the Android device probably won't work.
I'm not sure if any of that will help. For me, it just works. And I am not completely sure about some of the things you are doing to be able to suggest anything more.
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