Mathematical Model for SiC MOSFET C2M0080120D, transconductance and capacities
First off, most reputable manufacturers will give you models to their transistors, and Cree/Wolfspeed will give you LTSpice models. Modelling this piecewise in MatLab seems like a bigger pain than it's worth. That's just my two cents. On to your questions:


There's no difference between \$I_D\$ and \$I_{DS}\$
Transconductance is based on the derivative, not the average value. You can see that once the transistor is "on", the transconductance stays the same. While the (Vgs-Vt)/Id relationship may work for some devices, it does not work for most SiC MOSFETs, where the turn-on is much more gradual. 





The data for the ID/VGS curve appears to be for the 150°C Curve (see above). However, the paper does appear to be missing the piecewise equation that describes gm or Id relationship between (29) and (30) that is referenced in Fig. 5 as (30). If it's a journal article, those are frequently based on conference papers. Try looking for that.
I see the Ciss capacitance on the graph as 1100pF. I have no idea why it's different between the written part of the datasheet and the graph. It could be a version issue, as that particular part is relatively old and may have had a datasheet revision. Ciss isn't that important, compared to the gate charge if you're looking at input parameters.
Those look like matrix relationships for a simplified schematic (like the kind that a SPICE simulation does for you). It's based on the schematic that they have in Fig. 3d, and where it is in the simulation based on that. Look at their matrix definition immediately after Eq. (9) for what those may represent.