I am planning on making the above replacement.
The original fork is the composite fork with alloy steerer, however the replacement (from Giant) has a carbon steerer. Can I still use the same original headset?
As other have stated you should be able to use the old headset providing it is in good condition. If you are swapping forks due to an accident then replace the headset even if it feels smooth. Your alloy steerer likely used a "star fangled nut " to apply the headset preload. With a carbon steerer you need a expandable plug that is made for use with carbon steerers.
Giant make forks with two different steerer tube diameters. Overdrive is a standard 28.6 mm (1 1/8"), while Overdrive2 is an oversized 31.8 mm (1 1/4") steerer. Provided that the new fork has the same steerer tube diameter, you should be fine.
Giant's bike archive doesn't include the specs for the 2013 defy 4, but it has the 2013 defy 3 and defy 5 listed as both having the Overdrive steerer (28.6 mm), so it's safe to assume that yours will too. This means that if the fork is Overdrive, with a 28.6mm steerer, then it'll work fine.