Hi,
1. When you edit a shipping method in HikaShop you can see the "tax category" setting which allows you to select the tax category to be used for the shipping method fees. So as you can see, it can't dynamically use the tax category of the products.
What you can do to circumvent that is to create two warehouses, one for each tax rate and assign the warehouses to the products based on their tax category.
Then, create two shipping methods, one for each warehouse and select the corresponding tax category in each one.
That way, on the checkout, the products will be grouped per warehouse and the shipping methods corresponding to each group will use the same tax category as the products of that group for the shipping fee.
And the total shipping fee will be the sum of the shipping methods selected for each group.
2. You can use the minimum/maximum price restrictions of your shipping methods in order to display/remove them from the checkout.
Note however that if you have different groups of products, like explained on point 1, then the shipping methods restrictions will be based on the total of their group, not the total of the order.