List of workshop upgrades I shouldn’t buy in Iron Will mode

There are three main ways to get coal. The first two you'll get are the Coal Furnace and Deep Mining upgrades at the workshop. These will let you get small amounts of coal for each mine (0.003 per tick, 0.015 per second) and running smelter (base .005 per tick, .025 per second).

But, by far the best source of coal, is unlocking the Geology technology, and assigning Kittens to the Geologist job. Geologists are worth .015 coal per tick , equivalent to three smelters, or five mines. And kittens can improve at their jobs; a Master Geologist is earning about 0.026 coal per tick, which is over 0.1 coal per second each.

There are also quarries, which each give a +35% boost to mineral production, as well as another 0.015 coal per tick. They won't improve on their own the way Kittens improve their skills, but they still help a lot.

There are also several upgrades at the workshop related to coal which can help. Pyrolysis will increase coal gains by 20% from all sources. The Geodesy will increase Geologists' base coal per tick to 0.0225, making even a Dabbling Geologist worth four and a half Deep Mining mines. It also allows Geologists to produce a small amount of gold while they work.

No, there is nothing that need to be unlocked via Job Upgrades.

For some upgrades there are prerequisites, but most of the times that is just logical. For example you need to upgrade the Mineral Axe before you can get the Iron Axe. Before you can upgrade the Steel Axe, you'll have to have the Iron Axe. And so on. The same applies to barn/warehouse upgrades. An example of such a requirement and unlock is the following (from workshop.js):

   {
    name: "steelSaw",
    title: "Steel Saw",
    description: "Improve Lumber Mill efficiency by 20%",
    effects: {
        "lumberMillRatio" : 0.2
    },
    prices:[
        { name : "science", val: 52000 },
        { name : "steel", val: 750 }
    ],
    unlocked: false,
    researched: false,
    handler: function(game){
        game.workshop.get("titaniumSaw").unlocked = true;
    }

There is nothing like this to be found in the workshop for Job upgrades. So you are completely safe by omitting the Mineral Hoes et cetera.