Roof repair in Syracuse, and what winter here does to a roof

Usually not. Water enters at a penetration or a seam, runs along the underside of the deck or a rafter until something stops it, and only then drips. A roofer who goes straight to the point above the stain without tracing it back uphill is guessing. Expect them to be in the attic with a light before they price anything.

Icicles themselves are cosmetic. What they tell you is that snow on the roof is melting and refreezing at the eave, and that is the mechanism that pushes water back up under the shingles. The underlying cause is almost always heat escaping into the attic, which is why the durable fix is insulation and ventilation rather than knocking the ice off every winter.

For storm and hail damage, usually yes, and it is worth getting an independent assessment before the adjuster arrives. For age-related failure or repeated ice damming, generally no - policies cover sudden events, not wear or maintenance, and a denied claim still sits on your record.

For a genuinely isolated problem like one section of flashing, yes. For anything spread across the slope, no. Asphalt is typically rated 20-30 years and the failure curve steepens sharply at the end, so money spent patching in the last few years is rarely recovered.