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If you want better quality fish, then you should use something that adds to your fishing skill, such as the Dish O' The Sea. I'm pretty sure luck only effects the kind of fish you're getting and not the quality of the fish. The quality of the fish is determined by the size of the water and how far you cast in that water. You don't have to worry about the size of the water too much, more about how far you cast (The farther the better) Since casting distance is determined by your fishing skill, your fishing skill also determines what quality fish you are able to catch.
Combat isn't effected by luck and it's skill only adds to your health and how much exp you gain per kill.
I believe that the quality of foraging items are determined when they're first loaded in each week. I wouldn't waste a Lucky Lunch just to go foraging, use something that adds to your foraging skill, such as the Survival Burger.
Hopefully that wasn't too long and I answered your question properly lol.
Answer to complicated, used up 15 lucky lunches to go pick spring onions in one day.
lol :D
Farming levels DOES factor into how high the quality of your crops will be. The formula for checking if a crop will be gold quality when harvested according to the wiki is 0.2 * (farming level / 10) + 0.2 * (fertilizer level) * ((farming level + 2) / 12) + 0.01. Luck does not factor into this, and farming level provides a significant effect on the likelihood or harvesting gold crops. In my testing, harvesting the same 72 crops with a modified farming level of 13 yielded 9 basic , 22 silver, and 42 gold, compared to the 9 basic, 31 silver, and 33 gold the crops yielded without the bonus, so the buff is extremely noticeable.
In short it seems like you will always want to buff your farming stat when harvesting crops whenever possible, and you usually shouldn't bother with luck at all. Which I for one fully support as a design choice, because what is it you need to cook the Farmer's Lunch, the best farming buff food available? The humble parsnip, the very first crop you plant in the game, somehow making it still completely essential to harvest no matter how many years in you are and how many millions you're making. I think that's fantastic.