Is it normal to notice underneath the plastic barrier in crawlspace drops of water

The moisture barrier would generally go between the furring strips and the concrete and would be attached to the wall simply by attaching the furring strips to the concrete. If you wanted to really seal the area you could run a bead of adhesive sealant between the moisture barrier and the wood.

6mil plastic is a "vapor" barrier, not a "moisture" barrier. The difference is that a moisture barrier allows air to permeate it but prevents liquid water from going through. A vapor barrier prevents water AND air being transmitted through it. Moisture barrier is used between porous materials like concrete and wood to allow the various materials to acclimate to changing temperature and humidity without exchanging liquid water (the concrete will absorb a lot of liquid water from the ground, and then pass it into the wood which is a sponge for water, which will cause the wood to rot). If vapor barrier is used, moisture will be trapped on either side of the barrier, which can cause puddles of condensation on the moister and/or warmer side of the barrier.

So, to answer your question, you should probably use a product like Tyvek or TyPar, or an older material like roofing felt, between the furring strips and the concrete. For an installation like this, you can just cut strips of the barrier to prevent direct contact between the materials, and then simply place them behind the furring strips as you attach the strips to the wall.

"allow the dirt to breathe" doesn't even sound like an urban legend. Just sounds plain pointless. The entire point of the vapor barrier is to keep the moisture out of the crawlspace--so leave the barrier there. The dirt will be OK.