You can literally spend a lifetime researching the answers to your questions. Here is a very basic primer on the subject.

OP = open pollinated. This means that this is a population of related individual plants. These are produced by growing the parents in a large field and taking all the seed. Outcrossing, off types and variation is normal. Relative uniformity is maintained by roguing out off types.

Inbreds = are genetically mostly uniform (above 98%). These are used to produce OP's or F1 varieties . These are created by self-pollination, sib-pollination or double-haploid techniques.

F1 = A cross from two inbred lines. Most varieties grown from seed will produce fertile gametes with very few exceptions. Seed can be produced if you know how to. The reason F1's exist is because of heterosis or hybrid vigor. This makes the plants of some species (if bred corectly) healthier, more uniform, and produce more than OP or inbred lines. Saving seeds from an F1 will created and F2. F2 is the generation of maximum variation and the resulting plants will all be genetically unique.

Confusion of fertility of F1's: Sex expression in plants gets complicated. All male or all female plants are common in nature. Plants can also have separate male and female flowers on the same plant and all sorts of varietions. You can also change the sex expression of a plant with the application of some hormones. Plant breeders often take advantage of these diffences in variety development. Only varieties that have an odd number of chromosomes (like seedless watermelons) are truly sterile.

Saving seed - each species has its own techniques to successfully save seed. You'll have to research how to do it for everyone. Some are very easy to do like cantaloupes. Others can take specialized skill to save like lettuce.

Storing seed: Each species has its own storage requirements. You'll want to look up the requirements for each one. There are wide ranges of temperature and humidity. Some species can be stored for decades. Other species are only viable for a year.

Economics/risks of saving your own seed. Yhis gets extremely complicated fast as well. Seedborne diseases, pests, genetics etc all are parts to consider. There is no one size fits all answer.