What do I need to do to fix this?
It would depend- if you are fairly certain that the overflow issue cause is the fill valve then that needs to be addressed, as you already seem to know.
Is there anything I can do to fix the current valve?
Maybe- if you are fairly handy you could probably disassemble the float valve and repair or replace the washers inside, but it might be easier to just replace the valve. One trick would be to carefully bend the float-ball rod a bit more to the downward position. This causes a bit more pressure against the bibb washer when the level rises and might make it seal; this is a temporary/cheap-ass fix.
Or will I need to buy something like this?
Your link is to a new float valve, which is fairly inexpensive and easy to replace; I recommend it as long as you ensure that the diameter (the hole it fits into) and threaded connection size/type are the same or are adaptable to your existing equipment.
Do I need to make sure the diameter of the pipe matches up with a new
valve?
See previous comment above.
Does the height of the float need to be configurable on a new valve?
You can adjust the water level by using different length rods and bending them as needed.
When fitting a valve, will I need to use thread-seal tape?
Yes, on the threaded water connections to ensure a good seal.
Will I need to do anything to the bolts on the outside of the tank
(pictured above)?
If you replace the float valve you will need to ensure a seal between the valve and the tank. This is usually accomplished with a rubber gasket between the nut and the tank, if the old gasket is soft and crappy you made need to get a new one (the new float valve may come with one).
