Can this be install without your supplied faucet, so feeding from my cold water input and back into itself basically? And also to my refrigerator?

Dear Valued Customer, All under-the-counter Reverse Osmosis systems are not designed to dispense pure water from your regular sink faucet. It would require a dedicated drinking water faucet that we include with RO-Hi. As long you have good water pressure at the home, you can also supply the pure water from this unit to your refrigerator. Best Regards. APEC Product Specialist