Minecraft – Different number of dispensers activated depending on orientation of redstone dust

There is no item actually called “Redstone Dust”; this is a colloquial name for the  Redstone item, because it has the appearance of a pile or trail of powder. If this is what you have, then you already have the “dust” and you can use it as you have seen discussed.

If you have a stone-like block with red inclusions, then you have  Redstone Ore. This is not usually the case because it requires an pickaxe enchanted with Silk Touch to obtain without breaking the block. To obtain Redstone from Redstone Ore, simply place the block and then break it with an iron or diamond pickaxe that does not have the Silk Touch enchantment. (You can increase the quantity obtained by breaking it with a Fortune enchanted pickaxe.)

The most resource efficient way I can come up with to power a horizontal line of droppers is this

with the redstone line on the left being connected to a clock. This works because the repeaters strongly power the dropper they are pointing into causing the droppers to the left and right to be indirectly powered.

The vertical line of droppers is powered as described in this Youtube video, with a comparator being used to send a (inverted by a redstone torch) signal to a vertical stack of redstone torches powering repeaters pointing into every 2nd vertical dropper.

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With regards to the specific situation you are in the most resource efficient way to get the mobs from the exit of the spawner to the 'home' area would be to bring them up high enough using a mob elevator (a 1x1 tube filled with alternating water blocks and signs will cause the mobs to swim themselves up to the top) to a point where they can be dropped down the required number of blocks.

The only pitfall with this type of system is ensuring that the mobs fall a consistent distance everytime. All that is required to do this is to ensure that the water stream at the top of the elevator is set up like this

XXXXXXXXXX
X        X
X~~~~~S  X
XSXXXXX  X

(where ~ = water and S is a sign)

So that mobs get pushed into the drop shaft by other mobs pushing against them instead of directly by a water stream.