Claris ST-2300 shift/brake units compatible with V-brakes?
You are correct that the Altus rear derailleur will work with Claris 8 speed shifters. I think the front derailleur will work as well.

Regular V brakes will not work with the Claris brake levers. V-brakes require a different leverage ratio at the lever than road caliper and cantilever brakes. The 'mini' V-brakes apparently do with with road brake levers. 

The most important thing to be aware of is that replacing flat bars with drop bars is going to significantly affect the bike's cockpit geometry. Frames for drop bars generally have a shorter reach (horizontal distance between bottom bracket and top of head tube) than frames for flat bars. This is because the hand position on drop bars is further forward than flat bars. You may end up with bike that has an unridable hand position without a very short stem.

As stated in the comments, flat to drop bar conversions are problematic and not recommended. Apart from the the cockpit geometry issue shifter/brake units are a expensive component (compared to the cost of the whole bike). 

If you want the Giant to feel a bit faster, move your bars lower by dropping the stem down to the lowest position on the steerer (i.e., move the spacers from below it to above it), and possibly replace the stem with one a 10mm step longer. Stems are inexpensive and easy to swap out.