First, changing your stack height by moving spacers from below your stem to above your stem is mechanically very easy. About the only catch is that the stem acts as a locknut to secure your headset tension adjustment so when you reassemble the stem into the new lower position you have to reset the headset preload (the top cap on top of the stem does that) resecure the stem (making sure it's pointed straight forward and aligned with the bike) and tighten the stem bolts to secure everything. You'll want enough preload to make sure there's no headset slop (stand in front of the bike looking back towards the seat, hold the front brake closed with your right hand (only the front brake leave the back brake free) and gently rock the bike fore and aft to feel for headset play. Once the top cap is snug enough to remove play, make sure you haven't over tightened it such that it binds or feels tight when you steer the bars side to side. It's actually pretty easy with modern headsets and stems but you have to know what you're trying to achieve, nice smooth steering with no excess headset play.
BTW, you typically move the extra spacers from below the stem and reuse them above the stem. You'll need them to take up space above the stem if you don't want to cut your steering tube shorter - which you generally don't want to do unless you're absolutely certain your fit will never change and even then it's best to have a shop cut your steer tube to make sure it's cut perfectly straight.
I wouldn't have phrased dropping the stem in terms of weight on the bars. There's truth to that, but I think of it more in terms of comfortable fit and aerodynamics. If you're comfortable with a flatter lower back and want a bit of an advantage when going fast or pushing into a headwind and the lower position feels comfortable then lowering the bars by swapping spacers makes sense. Yeah it can change your weight distribution but the most noticable thing to most folks is bending over further, possibly reaching a bit further to the brake hoods and being lower, especially when riding in the drops. Try the position out for a while, once you know how to readjust your headset preload swapping back is a piece of cake if you don't like it or going even lower is a possibility if you have plenty of flexibility and tend to ride aggressively or push a lot of headwinds.
A lower position won't in general help your climbing, but unless you're pushing beyond your flexibility limits and moving towards an uncomfortable position it shouldn't really hurt your climbing either especially since you should typically be climbing with your hands up on the brake hoods or the flat bar tops.
Good luck,
-Dave
