Sorry for the delay in replying! In this period a lot of questions come up and in addition to the blog I have to find some time to look after my crops, however I hope to be useful to you.
The ideal fertilization
To understand which nutrient you should have the soil analyzed in the laboratory. If I were you, however, I would avoid the expense and try to cultivate the vegetable garden. Based on how the crops will come, I would decide to intervene on the way. There is nothing better than a nice fertilizer to give a new life to a soil earthworm humus, is a complete fertilizer, which therefore brings the various elements and microelements, I would limit myself to that. And if you want to do things right, start growing legumes, in this way you will fix nitrogen and further enrich the soil.
The period in which to fertilize depends above all on the type of fertilizer you will use. There are fast-release fertilizers, such as liquid fertilizers, which make the elements readily available to the plant but are easily washed by the rains, and long-release fertilizers, such as cornunghia, which release nutrients over time. My advice is to fertilize a couple of weeks before sowing or transplanting the vegetables you are going to grow.
As for the third and last question, here too it depends on the type of fertilizer. If you use mature fertilizers (humus, pelleted manure, manure left in heap, well composted humus, ...) you can also cultivate immediately. Personally, after a good basic fertilization I always prefer to let about ten days pass.
I hope I was useful, if you want to deepen these issues I suggest you read the pages related to garden land and to organic fertilization.