I will add one more thing, that will answer this part:
I my current game I have one camp and four plantations within four hexes of the Temple of Artemis so I should get 5 amenities for it that in that city. But as mentioned previously I get 4 extra amenities.
The way Temple of Artemis works isn't
"Give the City +X Amenity, X being the number of Camps, Pastures and Plantations within 4 tiles" but
"Each Camp, Pasture, and Plantation within 4 tiles of this wonder provides +1 Amenity."
The difference being that all Amenity generated by the Temple of Artemis will not necessarily go to the city which build it. Instead, the Amenity will go to the City which has one of the improvements listed in its border, if that improvement is within 4 tiles of a Temple of Artemis.
To add on pokiehl's text: it only applies to Amenity provided by Luxury Resources. Luxury Resources try to provide their Amenity to cities in need first. The Amenity provided by Temple of Artemis cannot go to other cities (unless you change the ownership of a Camp, Pasture and Plantation to another city).
In short:
- Temple of Artemis provides 1 Amenity to Camps, Pastures and Plantations within 4 tiles. Those Amenity will go to the city who own those improvements, and not the city which built the Temple of Artemis. That is why you only got +4 Amenity out of 5 affected tiles.
- Luxury Resources provide Amenity to Cities in need first, and try to have most cities with the same amount of Amenity. The following turn, the Amenity provided by Luxury Resources went to other cities within your Empire. That is why you went down to +1 Amenity: 3 of the Amenity you city enjoyed was from Luxury Resources, and went to other cities in need instead.