There is some water dripping behind the gutter, and I am thinking of installing a drip edge. From what I read a drip edge is installed under the ice water shield at the eaves. The ice water shield sticks to the surface. Should I be attempting to lift up the ice water shield to install the drip edge underneath? Is the ice water shield likely to rip when pulling it up? Or is it unlikely to stick back after? Any other issue with what I am proposing?
It is possible that your ice & water shield will peel up cleanly depending on age, climate, current temperature, sun exposure, etc. Try that first, otherwise...
Installing drip edge under the two layers of shingle that should be present will take care of 99% of the water flow. The small amount that may come down during ice dam conditions may run under the edge. You'll need to decide whether that's a concern in your case.
You could also install your edge, then seal it with a narrow strip of ice & water shield. It'll bond with the existing layer over time and seal against nearly all leaks.
The bottom line is that anything is likely to improve your current situation.
If there is water dripping out from underneath your shingles (on the ice & water), then there is a problem with the roof installation which probably won't be addressed by installing a drip edge. There should only be water getting under the shingles in the specific case where you have a large amount of ice buildup above the overhang of the roof (where it is cold), which causes meltwater from higher up the roof (above your attic, where it is warm) to pool on the roof. If this is happening during rainstorms, it is because you have a leak higher up the roof, and it probably should be looked at by a professional.
If the shingles do not overhang the roof edge enough (1.5 inches is standard) causing water to drip between your gutter and the fascia, the best DIY way to solve the problem would be to install something like this http://www.homedepot.com/p/Gibraltar-Building-Products-10-ft-White-Aluminum-Eave-Drip-Flashing-11368/202322782 in between the shingles and the ice & water.
It is only a couple of inches wide, so it should slide up under the shingles easily as there are not normally fasteners at the very bottom of the first row of shingles. Then you could fasten the lower part of the flashing to your fascia with a few coloured screws, rather than trying to lift up the shingles and fasten underneath them.
As I said, almost all of the water coming off the roof should travel on top of the shingles, so it is not necessary to put the drip edge under the ice and water shield, or try to seal the upper edge. If there is much water underneath the shingles, your roof is LEAKING, and that needs to be figured out first. Post some pics if you want a more detailed assessment.
When your roof is installed , typically the drip edge is installed and then the ice and water shield is over top of the drip edge. Your never gonna get that ice and water shield up without damaging your roof. What I recommend do is getting a cheap roll of aluminum coil at Home Depot, its very inexpensive. It's around 10 bucks for a decent roll. What I recommend doing is cutting strips of the coil about 4 inches wide, and take and stick them behind the existing drip edge, and then drill a hole through the existing drip edge and new coil piece and rivet them together. When you add the coil behind the existing drip edge make sure the bottom of the strip of coil is inside the gutter, and there is no space where water can leak out behind the gutter.