Why does standing water collect along the bottom edge of the roof

If you don't have a problem with water pooling any more, the real answer is probably to have a professional roofer look closely at the roof both on the inside and outside. The problem with a lot of roof leaks is that where the water shows up in the house isn't where it comes in. Water has an annoying tendency to run along internal beams and such, then come into the house far away. You can go over any edges with new tar and hope you catch it but in my experience this approach ends up being like playing whack-a-mole, you patch and wait for it to rain, then patch more and wait more, etc.

So I would say if you can do it yourself, look closely at the underside of the roof as well as the outside. If you don't find anything, a professional roofer is probably in order.

Sounds like either the gutters aren't secured tight enough, or you don't have a drip-edge flashing between the roof and gutter. The drip edge deflects water out and away from the house trim and into the gutter. You install it under the shingles, pulled out slightly from the trim, and secured with roofing cement.