How does the Store Word(SW) and Load Word(LW) instructions work, MIPS
$s and $t are register specifiers.  Your interpretation MEM[RegisterData(s) + offset] is basically correct.  $s means, essentially, "get the data out of register number s in the register file."  So if s in the instruction is the bit pattern 10001, then $s means: read the value out of register 17, add the sign-extended offset bits (the 16 bits represented by i in the encoding), and use that as the address to access the memory.  Similarly for the $t in the lw instruction.  It means write the data back into the register numbered t.

Your question about sign extension: look up two's complement.  The 16 bit two's complement number 0000 1111 1111 1111 is decimal +4095.  The 16 bit two's complement number 1000 1111 1111 1111 is decimal -28673.  The 32 bit two's complement number 0xffff8fff is also -28673.  The 32 bit two's complement number 0x80008fff is decimal -2147446785, which is clearly not what you want.