Try turning off the water at the main. Turn on all of the faucets in the house and let them drain. Turn the main back on. Might be some air in the line.
Loose washer
Yes, the freeze-proof water spigot is installed in the outdoor faucet.
No, if the current faucet doesn't have a sprayer, there is no place on the faucet to attach the sprayer. It would attach on the underneath side of the faucet in the center.
Depends on the type already installed
A little bit of water may seep into the basement.
To safely and effectively remove a faucet in your bathroom, start by turning off the water supply to the faucet. Use a wrench to loosen and remove the nuts and bolts holding the faucet in place. Carefully lift the faucet out of the sink or countertop. Clean the area where the faucet was installed before installing a new faucet.
To remove an old faucet from your sink, follow these steps: Turn off the water supply to the faucet. Disconnect the water supply lines from the faucet. Remove any mounting hardware holding the faucet in place. Lift the old faucet out of the sink. Clean the area where the old faucet was installed. Install the new faucet following the manufacturer's instructions.
My guess is your garbage disposer was also replaced and they forgot to knock out the plug before connecting the disposer hose/drain It sounds like you have a vacuum breaker on the drain for the dish washer. This is probably just a coincidence but your vacuum breaker needs to be replaced or repaired. It really has nothing to do with the faucet installation.
The steps involved in a bath faucet replacement are: 1. Turn off the water supply to the faucet. 2. Remove the old faucet by unscrewing it from the plumbing. 3. Clean the area where the new faucet will be installed. 4. Install the new faucet by following the manufacturer's instructions. 5. Turn the water supply back on and test the new faucet for leaks.
Installed incorrectly, or cheap/faulty faucet
Bathroom faucets are installed by pushing the faucet into the hole left by the old faucet. Before doing this, you'll probably want to apply plumber's tape or joint compound on the threads of the faucet stems and also apply caulk. Then you connect your water supply lines to the faucet stems and apply more plumber's tape to the shutoff valve threads. The last step is to tighten the coupling nuts to the faucet supply stems, taking care not to overtighten.
Turn the water off to the faucet. Remove the handle. The top of the faucet should unscrew to get to the ball valve. Replace the valve and gaskets then reassemble the faucet.