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Almost everything that you can do with digital logic you can do in an FPGA. So yes, all of the examples you cited you can place in an FPGA.
As for necessary knowledge, it really depends on what you're wanting to do with the FPGA. The examples you cited would require basic knowledge of digital design and probably one HDL, like verilog. Knowing how to read the schematics of the dev board would be helpfull too. There's also some learning curve with Quartus II, but the basics are not difficult.
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Ok I see, well I know how to read schematics, I design digital systems but using microcontrollers and I want to start using FPGAs because what I've read and what you just told me I could take one of this system and put everything in to one single chip and save a lot of space and money and also have a lot of more power to grow the design and functionality.
What I'm looking for is some basics to start with this journey, like a small not-very-powerful FPGA to experiment, to get started
I've heard about Nios II but I don't know if this is an FPGA or what and I don't want to buy a very powerful FPGA dev kit yet, I want to start with something basic to experiment, gain practice, knowledge and experience and then move to something bigger.
I've also seen the "BeMicro SDK", it's a small board but I don't know if this would be a good choice to begin.