Characteristics of a Laser Diode Below Threshold
It's normal for a laser diode to emit (very little) light below threshold current (cf. "Below the threshold current very little light is emitted by the laser structure."). I don't know, though, if there is a (simple) formula describing the P/I dependency in that operating region and also why one would care.

The fact that voltage stagnates is also no surprise: it's just the forward voltage of the laser diode which depends above some thresold only very little on current like in other diodes. So yes, the Shockley equation is a (more or less) suitable I/V equation. Such equations are always based on simplifications/presuppositions (simplified models of reality). Whether they are good enough for your purpose or not depends on your requirements.