I've been gardening for a few years and have had some similar problems and learned a lot. I also really dislike using chemicals. Each problem calls for a different approach, and sometimes the approach can create problems :).
With fungi - if you mean blight - I've been lucky enough not to have to deal with it. Try to keep plants healthy and leaves relatively dry. If you mean powdery mildew, I think it's a part of life to a certain degree and the best you can do is manage it. Try to keep leaves dry when watering, promote airflow around plants, and cut off old leaves that have a lot of mildew. They should still produce. If you mean fungus in the ground around plants, that's usually a good sign not a bad one.
With cabbage moths, the most effective protection is some type of fine netting/horticulture fleece over the plants to keep the moths from laying eggs, or applying BT (bacillus thuringiensis) periodically. It's a natural pesticide (a bacteria) that's extremely effective against caterpillars. I use it, and it's great, though I'm hoping to use less this year because it likely affects other insects in their larval stages as well (thinking ladybugs and lacewings).
With spider mites... sigh. The problem has seemed to get worse each year. This year I'm trying a more holistic and hands-off approach using compost exclusively (0 or very limited fertilizer use, even organic) to make sure plants are healthy enough to fight them off and produce despite their presence. Also covering soil with dense plantings and other smaller inter-crops to regulate soil temps, hold onto moisture, and encourage soil life. I'm also allowing a baseline population of them in some places (not that I've had a choice) to give predators a food supply so they'll move in and hopefully help keep things under control. The thinking here is a "soil-up" approach that focuses on soil, plant, and ecosystem health rather than simply attacking what may be a symptom of other problems I've created.
With fungi - if you mean blight - I've been lucky enough not to have to deal with it. Try to keep plants healthy and leaves relatively dry. If you mean powdery mildew, I think it's a part of life to a certain degree and the best you can do is manage it. Try to keep leaves dry when watering, promote airflow around plants, and cut off old leaves that have a lot of mildew. They should still produce. If you mean fungus in the ground around plants, that's usually a good sign not a bad one.
With cabbage moths, the most effective protection is some type of fine netting/horticulture fleece over the plants to keep the moths from laying eggs, or applying BT (bacillus thuringiensis) periodically. It's a natural pesticide (a bacteria) that's extremely effective against caterpillars. I use it, and it's great, though I'm hoping to use less this year because it likely affects other insects in their larval stages as well (thinking ladybugs and lacewings).
With spider mites... sigh. The problem has seemed to get worse each year. This year I'm trying a more holistic and hands-off approach using compost exclusively (0 or very limited fertilizer use, even organic) to make sure plants are healthy enough to fight them off and produce despite their presence. Also covering soil with dense plantings and other smaller inter-crops to regulate soil temps, hold onto moisture, and encourage soil life. I'm also allowing a baseline population of them in some places (not that I've had a choice) to give predators a food supply so they'll move in and hopefully help keep things under control. The thinking here is a "soil-up" approach that focuses on soil, plant, and ecosystem health rather than simply attacking what may be a symptom of other problems I've created.
Hope this is helpful!