How do you remove an under the sink water filter?

You don't need to buy another valve, unless you want to. You can simply cap the stub where the filter line is connected to the valve, hardware stores will supply the cap. Just disconnect the filter line and cap the valve where it came from.

You want to remove all of the tubing back to the connection on the valve.  You can buy a tubing cap that will screw on to the threads of the connection at the valve.  There is also a hole drilled in you sink waste water drain that is held in place with a clamp.  Undo the clamp and remove the waste water discharge tubing for the filter.  You can replace the sink downcomer that has the drilled hole in it or put a piece of aluminum air conditioning duct tape over the hole.  Everything is at the hardware store.  Take a piece of the tubing with the connector to the store so you can get the right size.   You might also want to think about keeping the RO system because it (or a new RO filter) should make high quality water.  If so, consider drilling another hole in the sink for the soap dispenser.  With a good hole saw, it is not hard.   You might also consider buying a new faucet that requires fewer holes, thus giving you an extra hole for the soap dispenser.

yes, you can cap the second output.  take photo and measure -- the guy at your local hardware will fix you up.

Yes, you need a new shut off valve. The pipes look bigger in the picture, but I am assuming you have a 1/2" supply. You would need a 1/2" compression fit with a 1/2" male pipe thread.   

I used a hospital grade water filter. There is a screw on cap on the filter canister that should only be hand tight. I didn't install under the sink, I installed it in my basement with clear access where well water enters my house. The plumbing code doesn't apply when you own the house and do the work. I had to buy a new hot water heater that had to be inspected by a licensed plumber because I didn't do the work, he made them come back and put a worthless pan under the hot water heater, I pointed out that they forgot to wipe the joints, the plumbing inspector said I keep telling them but they don't listen.

They make caps for plumbing so find the appropriate one. A picture would make it a lot easier to answer this question as shut off valves themselves have one outlet typically.