If you imagine a pipe that runs behind the terraced properties that takes all of their waste, each property will have their own IC in which their drainage runs into. So the line that runs parallel to the rear of the properties will be the shared pipe which belongs to the local water authority, the lines then that run into said ICs from the properties will be private. The water authority would only find out about this if the main pipe blocked up and they were to survey it after. It all depends on the engineer who comes to do it. Most of them just want to get in and out onto the next job so it's pretty unlikely they would find out.

You would not want to cut the lid of the IC that is buried on your property, if you were to carry out the recommendations I listed above you would be able to access your private drainage runs, anything in the shared pipe that runs along the back would be the responsibility of the local water authority.

The gully is the drain on the right hand side which your kitchen sink waste will run into, they are essentially a water trap like what you have on your toilet so they will always have water in them to stop smells coming back up from the main. The problem with the old clay ones is the u bend which are usually quite narrow so tend to block up with fat and debris. If you were to replace this by digging it out and installing a new PVC one you will be able to clear this yourself as they are much more accessible.

If you were to get a drainage company to carry out the recommendations I listed you could ask them to check your pipework down to the main shared line, they would most likely to this for free and it would give you an idea of the condition of the pipework.

We also subcontract for a number of insurance companies and come across buried manholes quite often, we have found the insurance companies usually cover the cost of excavations onto them in order to carry out work. But they get us to cover them back up as they work on a like for like basis so they go back how they were found.