What is exact specification of target machine ?

It is really bare ZX Spectrum 16K without any attached device except tape player and display.

What kind of programming language can I use ?

BASIC and assembler are probably most obvious choices but if you have a Basic compiler, Forth, C compiler than can target 16K model, it is fine to use them.

Is interrupt mode 2 allowed ?

Yes, for purpose of this Jam, consider granted that there is no devil device connected to bus and you can point I register to ROM and exploit one of the few usable vectors at locations xxFF. 
If you want to be really nice, either provide Jam version with interrupt table in ROM and final release version with table generated in RAM above 32768. Or you can give user option to choose between 16K and Compatibility mode right after game finishes loading and set up interrupt mode according his choice.