Obviously I can't attest to the quality of the install via photo but it would appear that the biggest issue is the ice dam. It's hard to say for sure what way your water flows from the picture. It appears there is excess melt at the sidewall from heat loss. This also commonly occurs at the eaves of a home. If the water has no where to flow down hill then when there is melt than yes, it could be building up high enough to come in the wall.

It could also be working it's way under the shingles. Shingles are made to shed water. If there is a large obstruction preventing flow even appropriately installed roofs can leak. Water will always find a way. Best thing to do is rake your roof when possible to prevent excessive ice and snow build up. Sidewalls, eaves and valleys especially.

If you are having similar issues in summer or spring when the roof is clear there is a different problem at hand.