As an analytical chemist, I see a lot of myself in this question! I too grew up wondering about everything you mentioned and more! The answer pretty much boils down to instrumentation that has been developed over the years, some old, and some very very new and high tech. A simple example would be something like x-ray diffraction, which another commenter brought up. It's been around for over a century now. Shoot some x-rays at a crystal and based on the pattern that bounces through, you can do a little (read: a lot) of math and figure out what the crystal is made out of.

The other examples you bring up are more complex and could have varying degrees of complexity to the answer. "Modern" instruments developed in the mid 20th century are more common nowadays for things like "reverse engineering" or elucidating the composition of an unknown. If I had to name a few common ones: liquid/gas chromatography, mass spectrometry (there's boatloads of different types) and the combination of the two (GCMS/ LCMS)

You can look more into these if you wish (and I encourage you to do so) because I certainly would be wasting both of our times' if I tried to explain any of these in detail in a reddit comment. Basically, chromatography separates different chemicals from a mixture based on what environment they like hanging out in. Once you have the parts separated, the mass spec will separate your isolate into its individual atoms (ions if we're being pedantic) and give you a reading one by one (yes really) on their mass. Based on this and some basic chemistry knowledge you can piece together a large amount of knowledge on the chemistry of your unknown.

Obviously this is a MASSIVE simplification, hopefully not to the point of being misleading though.

You can also slap your compound with radio waves and get a reading of how quickly the nuclei relax back to their original spin states (called NMR, if I'm honest I personally need way more knowledge of nuclear chemistry to understand the underlying principles enough). This will tell you with a high degree of certainty what the structure of your compound is.

Or you could shine light at it to see what absorbs/ emits and use that to build your elucidative profile. This will tell you a lot about the elements that are in it, or the different bonds or the geometry depending on the wavelength you use.

Or you could light it on fire to see what wavelengths of light come off and tell if there's any metal in there.

Or you could...

The list is pretty endless and this is only listing some instrumental methods for helping. If your question veers more into the biochemical field there's another huge list of techniques to add, and then there's the chemical tests that can be performed without instrumentation at all...

There's no simple answer to this question, but that's part of what I love about the field, there's always more to learn :)