Does New Jersey require an ice barrier under new shingles in Warren County?
Not under the state rule as published. NJ Bulletin 07-3 limits the ice-barrier requirement to areas where the January mean temperature is 25 F or less, which it says is Sussex County only. It is good practice elsewhere, but confirm the current edition with your construction office.
The bulletin dates from 2007 and was revised in 2015, so a more recent version may exist. A roofer who proposes a membrane at your eaves is following good practice in a snowy county, not a legal requirement in Warren County, so the choice is yours to weigh against cost.
The guide to ice dams in Warren County goes into the bulletin and the physics in more detail.
Is it safe to remove ice myself?
It is not. Roofs covered in ice and snow are slippery, ladders slide on frozen ground, and hammering at ice breaks shingles and gutters. Stay inside, protect the interior and call a roofer. Removing snow from the lower edge with a roof rake from the ground is a common safer measure.
Salt, de-icing products and heating cables are sold for the job. Some help in limited ways, and some stain or damage roofing, so ask the roofer before you try one.
The most valuable thing a homeowner can do is find where the house is losing heat. A thermal-image or an inspection of the attic is far more useful than a hammer.
When do leaks from ice dams show up?
Usually during or just after a thaw following snow, when meltwater pools behind the ice and pushes under the shingles. Stains often appear on ceilings near exterior walls, and spring thaws can reveal damage that was building quietly through the winter.
Because the cause is seasonal, the same spot may leak every year. If it does, that is a signal to look at the attic and not only the roof surface.
If you are not sure whether a stain is from ice or from a leak at flashing, read about roof leak repair.
What does a lasting fix for ice dams cost compared with removing the ice?
Removing ice is a short job with a modest scope, while air sealing, insulation and venting involve more work and sometimes other trades. The price of each depends on the attic and the roof, so the roofer should scope the removal and the permanent work separately and confirm both on site.
Separating the two lets you decide in stages. Many homeowners have the ice dealt with first, then book the attic work in the warmer months, when it is more comfortable to do and the roofer or insulation specialist has more time.
Ask for the attic recommendations in writing, with the reasoning. You are paying for a lasting answer, and the explanation is part of it.