Hi Katetea:
Thanks for keeping in touch.
I do not know about England plumbing, necessarily, but I do know that all fixtures need to be vented.. so when they tell you pipe doesn't have to go to roof and it is "sealed off"... they are wrong! Once again I point you to your local building/plumbing inspector...ASK HIM if vent pipe from kitchen sink can be "sealed off"...I am quite sure he will not like that.
The "vent" for dishwasher... do you think they mean a dishwasher AIR GAP fitting... some places, even here require air gap fittings to dishwashers... MOST places, however just ask plumbers to create a high loop so that water from sink can not drain back down and into dishwasher. This works just fine. Your sink looked high looped. Anyway, see what landlord says about this VENT for dishwasher... keep us posted.