Proper way to install Torch Down
For starters, torch down is a rubber product. It should NOT be installed over asphalt shingles. Swelling and contracting will eventually allow the asphalt shingles to wear a hole in the torchdown membrane from the underside. The same applies for rubber membranes.

You are correct about the drip edge needing to be under the roofing material. This applies to ALL roof coverings.

Your roofer should have mentioned the soft spot prior to covering it up. That's complete irresponsibility. Roofers typically don't repair facia but it shouldn't be beyond a decent roofers skill set.

As for what the rafter you need repaired... Unless there's access from below, a portion of your new touchdown will need to be removed, along with the decking above. Unless the rafter is totally rotted out, I'd scab another 2x to the side of the rafter in question instead of removing it completely.

Additional details per OP questions:

The recommended overhang for torch down is 1/2".

For roof pitches below 12/4 torch down or rubber are my recommendation.

To my knowledge, it's not standard practice to double up material around the drip edge.

Since I'm unsure what your rafters are, I used a generic term "2x" ( pronounced 2 by ), which can mean anything from 2x2 to 2x12. 

Yes, if there's any bad decking it should be replaced, but it can be spot fixed from rafter to rafter.

Additional Info

Decking is a generic term... Meaning your roof deck ( sheathing ). Spot fixing means cutting out the bad decking from rafter center to rafter center.

I've never heard of Timbor. I ALWAYS replace rotten/dry rotted decking.

To repair the damaged rafter, screw or nail a new rafter to the side of the bad one.