Is it possible to simulate hardware of MIPS architecture computer defined using Verilog/VHDL?
Yes, this is most definitely possible.  This is what Verilog and VHDL were designed for in the first place.  There are multiple open source Verilog and VHDL simulators available, including Icarus Verilog, cver, GHDL, and probably quite a few more.  You may need to install a waveform viewer such as gtkwave as well to look at the simulation traces.  

Most of the FPGA toolchains will also include a free simulator as well.  Xilinx ISE comes bundled with isim and can be downloaded for free, though you pay for it in hard drive space (~10 GB).  I believe their newer Vivado environment is similar in both bundled features and hard drive footprint, though I'm not familiar with it as I use mainly Xilinx 6 series devices which require ISE.  Altera's Quartus software includes a free version of Modelsim and can also be downloaded for free, but it's also about 10 GB.  The open source simulators are three orders of magnitude smaller (~10 MB).