vat on invoice

If you ant to display the shipping without taxes on the invoice, you need to edit the file "invoice" of the view "checkout" via the menu Display->Views and change the code:

echo $this->currencyHelper->format($this->order->order_shipping_price,$this->order->order_currency_id);

by:

echo $this->currencyHelper->format($this->order->order_shipping_price-$this->order->order_shipping_tax,$this->order->order_currency_id);


For the address details, it's because you created new address fields and you don't use the normal fields.
So, you need to edit the files address_template (there are several of them) via the menu Display->Views to use tags corresponding to your new address fields

For the shipping methods selection, you can edit the file shipping of the view checkout via Display->Views and remove the code:
if(bccomp($taxes,0,5)==0){
$prices_taxes = 0;
}

For the shipping fee in the cart display, you can edit the file cart of the same view and change:
if(bccomp($taxes,0,5)==0){
echo $this->currencyHelper->format(@$this->shipping->shipping_price,$this->shipping->shipping_currency_id);
}else{
echo $this->currencyHelper->format(@$this->shipping->shipping_price_with_tax,$this->shipping->shipping_currency_id);
}

to:
echo $this->currencyHelper->format(@$this->shipping->shipping_price,$this->shipping->shipping_currency_id);