Vented soffit under bay window
Vented soffit below the window will allow air in, and will reduce the effectiveness of insulation. Other than that, it wouldn't cause a problem, but you're already concerned about the cold near the window. 

You should focus on how the water gets in, and prevent that from happening in the first place. It should be fairly easy for your contractor to tell what's been happening once the siding is off. It's hard to tell how things were flashed around the window from your pictures, but it doesn't look like how I'd have done it. 

When you have an ice dam, the real concern is water being pushed back up under the shingles. If that's happening, water could be dripping down the internal structure, and vented soffit below the window wouldn't help anyhow. Again, it should hopefully be easy to tell once the siding is off. If you haven't already, mention the water in the house to your contractor, and show them where you've seen it.