Programming FPGA chip

You can't program the FPGA and put it on another board, as the configuration is volatile and will disappear when you turn the power off. If you want to design your own board, you'll have to add a configuration device such as an EPCS flash memory. Add also a JTAG interface, you'll need it.

For the language, you can directly integrate vhdl code in your FPGA design. For the C++ code you will need to add a CPU, such as the Nios. But you will need to recompile the source code, you can't directly use a dll.

You can't use matlab code directly. Depending on the algorithm, you could rewrite it either as vhdl or C/C++ code and add it to your design. Some tools such as DSP builder can help you convert a Simulink project to VHDL but its capabilities may be a bit limited, depending on what you want to do.