I'd certainly think so. I almost did. I initially learned Java on my own. It's good to know at least one full programming language before diving into embedded programming, I think.

Then got into Arduino programming. There are tutorials for that online. Try communicating with other Chips like a Shift-Register, then something that uses a standard serial protocol (ex. I²C).

When you feel like you have a good grasp of the basics, I recommend getting a development board and doing the same there.

Most Microcontrollers (ARM Cortex Processors at least) are pretty similar: You get a datasheet and a User's Manual. The User's Manual describes a bunch memory addresses, which control the built-in peripherals. There are excellent descriptions of what value will give you what result, but it can be a bit daunting to get your head around it at first.

I recommend a chip that has a so called "Board support Package". I have experience with LPCOPEN. This will make things a bit easier, as you don't have to figure out each register address for each thing and can instead use functions like "Chip_TIMER_Enable(timer_t timer)".

There is a lot more to it, but once you get started, you usually always see the next step.

Neat, thanks for the detailed answer!

I self-taught enough to design and build a working PCB (that passed a bunch of certification testing), including writing programs for an ARM Cortex microcontroller. Took about six months of learning and then a couple months of design/implementation to get a product that can be sold and have the fundamentals in place to make similar kinds of products much more rapidly.

It's very doable with dedicated study and I'd argue it's one of the best ways to get your design ability to rise to the level of being able to build something from scratch, without reference to other code / searching google for answers.

Yes, easily. Though you may find it will pay much less than webdev, even for highly complicated work with Linux capable SoCs based board designs.