Variant Display Order

The values in characteristics dropdown will be orderable by id ( so from the oldest you added to the newest ) or by alphabetic name from next version.
The variants array ordering is irrevelant to the display since variants are not directly displayed in HikaShop's front end.

You're right, by default the characteristics are ordered by name. We added a new option to select whether to display them order by id or by name with next version which we will publish tomorrow.

The dimensions in the product information are used for shipping volume restrictions. They are not currently displayed on the product page.
You can edit the product page code to display them:
go to the menu Display->Views and edit the file show of the view product. There, you can add code to display the dimensions. For example:

echo 'Width: '.$this->element->product_width.$this->element->product_dimension_unit;

If you add a new characteristic to a product for which you already changed the variants, the system might not be able to decide how to generate the missing variants and change the existing variants to add that new characteristic.
Thus, you need to click on the "manage variants" button and add the variants of your product yourself.
Otherwise, just delete the characteristics. Save once. Make sure that the variants are all deleted. Add back your characteristics and the system should properly regenerate all the variants.

When you create a characteristic via the Products->Caracteristics menu, you need to set some values in it. Otherwise, you will get these errors as the system is not able to generate the proper variants because you have a characteristic without values. Please make sure that you have values for the characteristic "Shipped To"

Could you give us an access to your back end so that we could have a look at how you configured all that ?

So the variants were not regenerated as you already had variants.

What I did:
1/ Go on the product page. Click on the "manage variants" button. Select all the variants and click on delete
2/ Go back to the product page. delete the characteristics and click on "apply"
3/ Add the characteristics again and click on "apply"

The two products you linked to are now displayed properly on the front end