>How relevant is embedded systems programming today?
Embedded systems are everywhere, from your dishwasher to cheap LED lighting strips to smart TVs. The challenges and workflows have changed over the years though. There is still room for the old style (ie. I read a datasheet and write to registers on the vendor IDE) which is geared more towards an EE or hardware engineer and the new style (I take an RTOS and import all my libraries that I need and just add some glue code to get things to work) which is geared more towards a CS major. We can assume the trend will be moving toward the new style though, and along with that come a 1:10 ratio of software programmers to embedded programmers. Embedded has and will remain a niche but still relevant and needed.
>What is the future of its popularity? Will it start to become less or more relevant?
Nobody can predict the future, the best thing to do is to learn the most up-to-date techniques and tools to assist in development. Many skills of an embedded engineer are transferable. Since the field changes with time (as most forms of engineering do), it at least gives you a way to compete with the experienced players.
>Is it a stable career choice? How well does it pay?
Probably still pretty safe, especially with the world being more accepting of home office. Depending on if you can afford a setup of your own it might even give an advantage to the standard web developer. If you have a small lab set up in your home office you can likely charge a bit more or be more valuable. There is that barrier, though it is becoming less and less.
Pay is a bit less than software developers but honestly, it should be enough for a comfortable middle-class lifestyle. If money is a concern it probably better to learn how to ask for more money rather than select a field the has slightly more than the average salary.
>What kinds of things are embedded systems developed for? (what kind of systems will mostly I be working with)
That is a big question, it could be for something very simple, like a temperature sensor) or something complex like a temperature sensor). I find there are 3 categories:
You should find out what you enjoy most, there is room for development in all those areas you just have to look for it.
>What are the responsibilities I will have if I become an embedded systems programmer?
That depends, in a startup, you will probably have to deal with the hardware design (at least MCU selection and pinout assignments), the firmware, the development process, the connecting software for viewing or configuring your device, the testing and manufacturing jigs. In a larger company you probably have to learn how to use the senior developer's library and how to follow the workflow, it will be less demanding... That being said I only have experience in startup style and a bit of academia. Try to work with at least one other embedded engineer or on open source projects, one-man shows tend to lead to cognitive dissonance regarding one's abilities.
I have been in the field for 7 years now or so, but this is what I have wanted to do since high school. A few other opinions:
Hope this helps!