I found tons of pointers online for solid wood, but not recommendations for engineered wood. What kind of gap should I leave for the transition to tile, and how much between the floor and drywall?

Thanks.

That information should be available with the general instructions provided by the manufacturer.
It will depend to a degree on whether you are floating this floor or gluing it down.

There seem to be no instructions provided and I'm gluing them down. I'm not so worried about the gap around the wall but I am about the gap between the wood and tiles (which are about 1/8 inch lower than the wood). Do I need to leave room for a transition piece or do they float on top?

Well ideally, you would use an off-set t-mould (one side is lower than the other) to accomodate the junction between the two floors. The stem of the t-mould is generally about 1/2 inch wide and the overall width of the t-mould is about 1 1/2 to 2 inches. Leave about 7/8th inch gap between the floors, install the t-mould so that the stem is up tight against the tile. This will give you about 3/8th of an inch of expansion tolerance for your hardwood. Should be sufficient for all but the most extreme circumstances.