I'm certainly not an expert but if the plant blooms and then goes to seed, the natural order would be that the seed drops to the ground and may sprout in the fall or spring.

Yes, I agree. Go ahead and plant as many seeds as you want. But be aware that some hybrid (produced by cross breeding) plants do not produce seed. For instance, my Lord Baltimore hibiscus doesn't. And if the hybrid plant does produce seed the flowers may not be the same as the parent. But who knows -- they might be better! ... :-)

However, native and other plants like Texas star hibiscus will come true from seed.

Some seeds seem to have a built in 'season timer' (for lack of a better word), and wait until for the optimal season to sprout, but most do not and will come up when planted.

So just have fun and plant!

Of all the plants I've grown so far, I noticed that zinnias seem to naturally seed quickly enough that a new round of plants might come up in the same season, without human intervention.

Some seeds take time to mature and dry, but some need a stratification of cold weather. You can plant the seeds or simulate cold weather by putting them in the fridge.

I think that is why many plants, on their own without us helping, tend to not come up again until the following spring. That and probably with thin growth from other plants in early spring, a seedling has a chance to compete.

Now, that is just talking about seeds. What you have are many plants that you can take cuttings from, root them, and then plant them out as new plants. For that, all you need is a proper amount of growth from the mother plant, and then select your cutting. Other than cuttings that fail to root, you can effectively make as many new plants as you want from all the cuttings you want to take. The benefit of this is that you can do it this season with the new plant growth.

For example, the Rose of Sharon is quite an easy plant to make more of, not only from seeds that are happy to drop and make new plants next year, but also from cuttings that you do this year.

For some video tutorials, just type in "how to propagate plant-name" or "how to propagate plant-name from cuttings" in Youtube.Bee balm will usually reseed its current area pretty readily, as long as it's not covered in mulch. So you'll just need to collect its seeds (or next year, transplant seedlings) to get them to other areas.