Hardwood flooring: transitions and expansion gaps
The metal edge you see is a Schluter trim that installs under the tile. 

The expansion rate changes radically based on the climate you're in--drier climates tend to shrink more. Regardless of where you live you should have done moisture tests on the wood, the subfloor and the air in the home. Relative humidity is ideal between 30/ 55% in most cases and for most pre-finished hardwood floors. They do sometimes vary so read the instructions. 

For 3" wide and below the moisture readings should be with in 4% of each other; 3" and above should be within 2% of each other. This is pretty normal but again read instructions--some manufacturers vary. 

With all that said if its 3/4" thick, leave 3/4" expansion on the perimeter. this doesn't work well for baseboard install so cutting the drywall up 7/8" allows you to set wood 1/4" away from wall and any base will cover. 

Stair nose is part of the floor, so install it tight. 1/8" gap at tile to hardwood is plenty. Use a color match sealant to fill that gap for movement.