Not sure what you mean by "mode 1" and "mode 2"... do you mean "basic" and "enhanced" mode ?
In Enhanced mode both joysticks are mapped. See the mappings here:
In theory, if it can be done with a Joystick, then it can be done with the Unijoysticle in Enhanced Mode.
- if you need to press"space", you can do it by pressing "fire" in J1, which is button "B" (at least on Xbox One)
- Of course, you cannot press F7 or CTRL with the Unijoysticle.
It is not clear to me what is not mapped... but again, I'm not a Turrican player.