1. That's because you need to configure them properly. I've done it for you for the women and men menu items. So just compare with the others and do the same.
Thank you Nicolas for helping with this. It's working fine now.
2. That's something you should see with the support of the module you're using... I've done it for you by simply entering the id of a category listing menu in the itemid option of the module and it's now working.
This one I see no change - it is still defaulting to the women's category as indicated above. It seems to only do this whenever the sub-categories have the same name. It does this to the "Clothing" and "Shoes" sub-categories when you click on them in the MEN's menu they default to the WOMEN's menu.
This is also odd because it worked fine on the old site so not sure what the issue could be. I spent approx a week trying to figure this out with the owner of the module as I thought it would the module that was causing the issue and have no luck so far

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3. The Category seems to be stuck on Page 2. Please use the "Women" category to test what I am trying to explain as it is the only category with the pagination at the bottom.
If you click on the menu "Women", and look at the bottom of the page you will see that there are two pages. If I click on page 2, then click on the menu "Women" after that, you will notice that it stays on page 2 - should it not default to page 1? And no matter where on the site I go when I come back to that menu it always takes me to page 2 rather than page 1.
I thought it was an issue with my browser but I tried it in other browsers and the same thing happens.
Any idea why that is happening?
Thank you!