He should probably be hand stripped, not clipped, brushed well and often vs shampooed - you CAN wash him if he gets into something nasty, he won't explode, but proper stripping and brushing is far better for his skin. This may well help with his itchy/ponginess, if he has a build up of dead hair and undercoat, that will not help him.
If he pongs then its likely his food.
Clip a light weight chain to a ring bolt by the door (it can be removed and the hole filled when you leave), clip him on it as you open the door, then you can take your eye off the ball for a minute if necessary. Or keep a long line by the door and stand there with him every time - thats on you for renting an unsuitable property.
Fleas - treat the house regularly with something long lasting, the vet will know which is the best (Acclaim or Indorex usually) you DO have to use it properly, whole house, all down the nooks and crannies, crevices along the floors, under sofas etc. I see so many issues when people treat where the dog goes, forgetting fleas jump, walk, and hitch a ride on humans.
Then you only need to treat the dog himself if you actually see evidence of fleas -thats live fleas or flea dirt (its a dark brownish black dirt on the comb that blooms into a reddish brown stain on a wet white tissue).
Likely his itching is to do with whatever is disagreeing and causing his pong, fix one, you'll fix the other.
Walk him more -more enrichment - safe hired fields for hurtling around are great.
Sounds like some of your neighbours assume lurcher = traveller = dodgy. Ignore them, they are twats.