Answers

I would rather think your gutter needs a good sealant. If your soffit is sagging, you have some structural issues.Most likely nothing to do with the gutter. 

The most important thing would to review its structual integrity. This can be accomplished by removing some of the soffit panels and taking a peak inside. It could be a a rafter tail is comprimised or broken. Therefore, it would need to be fixed by lifting and sistering. (placing dimensional lumber beside the rafter tail and fastening together). Another fix would be to install a column to support the roof. Installing the column would require a stable footing below, (a concrete pad) and lifting the roof to the proper slope and affixing the colum to the sub-fascia board on the eave. The column method would provide the best support over time and combat heavy snow loads, however a "post" would be in the walk path most likely.