Run a strip of masking tape on the wall where the molding will cover it up, but where your nails will hit it.
I hammer nails into old plaster all the time by putting a piece of masking tape on the wall and hammering into it. It holds the plaster together while the nail hits it and can easily be ripped off (gently) after I'm done hitting it. In your application, you can just leave it hidden behind the boards!
Use masking tape because as, for example, duct or painters tape dry they grab onto the wall harder. I have pulled large chunks of plaster out by trying to remove old duct tape. I have successfully pulled fresh duct tape off of plaster walls, but I don't recommend trying.
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
I understand not wanting to risk damaging that original trim, but it appears the piece is only a few inches long and not solidly attached. It's probably easiest to carefully remove it, use plaster and/or drywall to fill in the area behind, then reattach the short piece of trim. If you damage it slightly, you can fill dents and cracks with wood putty, then just sand and paint.