Excellent pics and welcome to the site. You have a classic situation. Replace the faucet and the supply lines ... just do it. You are not the first guy to attempt a small repair that escalated into a more major one.
Turn off the water at the angle stops (small valves). If they don't stop the water, you will have to replace them (which is a good idea because they do corrode over time). They are standard compression fittings. You will need new supply lines and I recommend the braided version because they alomst never fail or burst.
The next adventure will involved removing the nut which holds the faucet in place. Once thet nut and the plate are removed on both sides, the supply line pipes can be detached at the bottom. (Have a big bowl and towel available for the run off) Now you can lift the old faucet out and drop the new one into the two holes ... but attach the supply lines first and push them down through the holes (if they will fit). If they do not fit, slip the faucet in place and tighten the hold down nuts/plates from underneath. Some faucets require a "basin wrench" to tighten the new nuts from underneath ... sometimes you can do it by hand (if you are flexible and strong). I typically put a bead of kitchen/bath caulk under the faucet's bottom plate then set the faucet onto the counter top. The manufacturers don't require the caulk but I like to make sure there are no possible leaks.
Tighten everything to "feel." Turn on water and test for leaks. TUrn connections tighter as necessary but do not overtighten (rookie mistake)