porkpal said:My vegetable garden usually turns into a jungle, so I am not a good one to offer advice.
At least your plants grow. Should I leave anything in my brother's care it would turn into a dead zone in under a week.
To get back to the original topic, yes, many fungicides and insecticides can be mixed but it's always better to prepare a schedule ahead.
For example tomatos and eggplants should be sprayed with copper every fortnight, Garden sage or Dog rose should be planted as a "telltale plant": when it gets the first spots of powdery mildew, it's time to spray all susceptible plants with lime sulfur. And so on.
Generally speaking pests are treated on a strict "if needed" basis: if they present themselves, you kill them. If they don't show up, no need to intervene. That's as easy as that. That's why large commercial operations have all sorts of chromatic and hormone-based traps around them: by monitoring them commercial growers know exactly if and when to treat their crops for pests. Since we only grow relatively small patches and gardens, we can check on our crops regularly without the need for traps.
One final note: home-made remedies cause more troubles to plants than they solve. For example many people use ordinary soap instead of insecticidal (potassium) soap and end up doing nothing about their pest problem and causing their plants problems by spraying in the hottest hours of the days and causing leaf scorch. Neem oil is a particular polemic idol of mine so if you hear me rant and rave about it don't worry: while it has its (limited) uses it's sold as a panacea and people end up spraying it for everything way too often causing, you guessed it, leaf scorch!