I have a stench pipe that I want to get rid of because it is :

1. asbestos
2. ugly
3. damaged

My questions are:

1. do I have to have one?
2. if the answer to the above is "yes", is there a more unobtrusive alternative e.g. some sort of valve near the ground?
3. would it be permissible to run a washing machine outlet pipe into the same pipe

The property is a detached bungalow with similar properties either side.

1. Do I have to have one? (Stench pipe)

First off it's called a vent and, yes, you have to have vents on every fixture that has a trap for your system to work properly.

2. if the answer to the above is "yes", is there a more unobtrusive alternative e.g. some sort of valve near the ground?

Not if this is the main vent. If this is a outside vent then you could convert to PVC and paint it the same color as the house so it would blend in.

3. would it be permissible to run a washing machine outlet pipe into the same pipe

Only if it were the top most fixture and only if the fixture under it were vented. However, If you were to vent the washer you may connect to the vent at any point you wish.

In short. Vents are just as important to your drainage system as your drain lines. Regards, Tom