Behind the handle for the water shutoff, will be a "packing" nut.
An adjustable spanner can be used to tighten this.
Or a cresent wrench, or pliers.
If that doesn't stop the leak or dripping, you may need to replace the shutoff.
For that, you will need to shut off the main for the home.
The plastic tub is there to prevent too much water from flushing out.
If the toilet has been working fine, I would leave it.
If you remove it, you will need a 1.6 gallon flapper to prevent the entire tank from draining.