Yes, VAT can be applicable on Cart Page, Let us know if you need to apply VAT on Cart Page.
Yes, VAT Exempt can be set as filterable attribute.
Yes, It can be exempted.
Yes, VAT Exempt supports product with custom price.
Yes, VAT Exempt Extension is applicable for orders placed from admin.
Yes, it works smoothly with SUPEE-6788 security patch.
Community version: 1.7, 1.8, 1.9, 22.214.171.124
Yes, it support.