1)Yes, today most of the deep learning model can be implemented on FPGA provided you are able to generate synthesisable RTL code which can be dumped on FPGA.
2)Yes, simulink models support RTL code generation and verification by using products HDL coder and HDL verifier respectively. If you can generate your logic through system level modelling on simulink through HDL compatible blocks then easily they can be used to generate RTL synthesisable compatible to both altera FPGA from INTEL and xilinx FPGA Boards.
3)Most of the boards are programmable with matlab.
Hope it helps !