If you can mark the shims at the face of the cabinet, remove them temporarily, cut them slightly shorter than the mark, use a black marker to "paint the visible sides of the shims and set them back in about a 1/8" behind the face of the cabinet. Using a tube of clear water cleanup caulk, with the tip cut really small, say about 1/8" at the tip, draw a line around the bottom of the cabinet, filling the gap. Wipe the excess flush with the edges of the gap, leaving nothing on the adjacent surfaces of the cabinet or tile, the residue will show up in the glare from the lights in the bathroom.
FWIW, sealing the bottom with caulk may be good if the sink, waterlines or drain does not leak in the years to come. I would do a little here and there with the clear caulk at the escutcheons to keep the potential leaks in visible places, rather than running under a cabinet and collecting there. If the toilet floods, it is a good thing to caulk there, it is a bad thing if the drain or something else at the inside of the cabinet leaks. That is why I suggest the caulk here and there wiped with a damp sponge or cloth to keep the caulk in the joints only, and force a leak to show itself.
I see you have no base there yet, I would caulk the gap at the wall to the floor before you install the base. Just my 2 cents worth...
Before you attempt to repair it, make sure that the cause of the ice dam has been fixed. You may need to insulate the ceiling, and/or add a styrofoam vent baffle to the inside of the roof to prevent that area from getting too warm. Ice and water shield installed on the roof probably would have also prevented this from happening.
The repair will depend on how long the wall was wet and the extent of the damage. From what I am seeing, you should be able to fix this damage without needing to pull the whole wall apart.
Let the wall completely dry out and then check for signs of mold growth. A dehumidifier is not going to dry out the wall for you. You will also need fans aimed at the wall to circulate the air. Industrial blowers will do the best job, but a household box fan on high will also do.
Mold may be on the surface of the plaster, and it could also be under your trim. Use white vinegar to clean off the areas affected by mold and then use a mold killing primer such as Killz on all of the damaged areas.
Once the wall has been primed, then you can repair the plaster. You will need to get a bag of plaster of paris and a 5 gallon bucket of drywall joint compound. Mix the plaster with water per the instructions and mix in joint compound in a 50/50 ratio. Trowel on the mixture and blend the repair into the existing wall. The advantage of mixing the two is that you can use thicker coats of plaster without having it crack. It will also dry much faster and shrink less. The joint compound will make the repaired area harder and will bond better with the wall. Over time plaster dries out and becomes brittle. The gypsum in the joint compound will help prevent the plaster from deteriorating. In the old days, they would mix in chopped up animal hair (usually horse hair) to strengthen it instead.
You may need to apply more than one coat. You should also sand lightly between coats and feather the repair into the wall as best as you can. Any imperfections will be easily seen once the wall has been painted.
Once you are satisfied with the repair, then you can prime the wall. You can use any kind of primer that is compatible with the color coat. I prefer using primer that is rated for interior and exterior use since it is less likely to peel when you put the final coat on.
Repaint the wall and trim. If you can, repaint the entire wall at once so there isn't any differences in the sheen. It may look different if you only paint the damaged part.
Once the wall is painted, you shouldn't be able to tell that there was ever any damage there.
Best Answer
That does not even qualify as a hole as the back of the drywall is not broken. Just push some drywall mud, vinyl Spackle or caulking into it and call it good. you don't even need to make it pretty, just make sure it flush with the wall so it does not interfere with the vanity.