Hi Welcome to JustAnswer. My first response will follow shortly. Please feel free to follow up if anything is not clear

The issue is that the name is XXXXXXX XX two respects, being similar in sound and in the design. Trade me are taking a conservative view because they are concerned about a lawsuit from Adidas

The problem with trademark cases is that they are decided on issues of whether the marks and designs and names which you are using are confusingly similar to those of the registered trademark owner. Your names sound similar and if you are using the three bones, this adds to the similarity. You may have difficulty getting Trade Me to accept the advertising from you because of the fear of similarity. But until Adidas try to restrain you, you can continue trading. I am confident that if they think you are breaching the trademark, they will write you a warning letter before they take action. It is not so very similar that it is obvious.

It is interesting that it is Trade Me who are taking steps, and not Adidas. Of course Adidas may be applying pressure to Trade Me. You need to find out if that is the position, because if Adidas are not taking any steps to protect their trademark, then there seems no reason why Trade Me should be stopping you either.

When it comes down to it, Adidas is about running shoes and athletic equipment. when I look at your site, it does look similar even with the bones. In my view, Adidas could succeed in restraining you from selling this product. But before you stop selling it, you should enquire from Trade Me whether Adidas has in fact intervened.

In my view, you would need to work on changing the design because the words sound the same, and the three bones look similar, to such an extent that I believe Adidas could succeed in restraining you from selling.

I think you need to come up with a different design. You may be able to get away with Adidog

on its own, with the combination with the three stripes, even made to look like bones, make this look too similar. I would avoid the bones and the diagram for a start.