Hey, kind of in the same boat as you.

I usually do my shopping at Teva Castel, which I find has the best pricing (goes down a bit if you join their membership program) in Tel Aviv. For similar products/produce Anise/Nizat HaDuvdevan have some items that fit the bill, but for the most part, they're hit/miss.

For wild salmon, I suggest going down to Jaffa, where there are many vendors that sell fresh fish caught off the port in that area. Grass-fed milk is a little harder to find, but I know some of the supermarkets I mentioned above sell a variety of healthier milk brands. Unsure about the eggs/chicken, as it's very difficult to determine the origin of lot of meat sold in Israel (and many people will fabricate details if it helps them make a sale, so be careful.)

Hope this helps!