This user has no pubic description
Tbh I’d just sand down the edges of that and throw on another sheet of veneer. Other than that, your idea of wood fill and paint would probably work (I’d look for a touch up marker in the same wood type), but seems like a lot of effort for particle board and wood veneer
With no offense intended (because the fault is on the cabinet maker), that’s one of the flimsiest looking designs I’ve seen. In most cases, the face is just a plate attached to the cabinet itself.
As for fixing it, it looks like it was just stapled through the sides. I’d pull the staples and clean up the bits where the face ripped off. Maybe try filling in those parts of the face plate with some wood fill or bondo. Then you can run some glue along the cabinet and glue the face back on. Maybe run some L-brackets and nails/screws into it, too, if it still feels flimsy.
Do you know where the water is getting in at? I see that it’s coming out from the grout. If I understand the situation correctly, all that re-sealing that hole on the outside will do is make it so that the water sits inside the framing. I wouldn’t plug up the outlet hole until you’ve made sure that no more water will get in there.
ETA: As far as sealing it goes, I’d personally just smear a bead of silicone across wherever the water is getting in/out from. Grout can be a bit tricky to patch and blend, in my experience.
Spray foam. But your landlord should be fixing this, not you
This is TempleOS erasure