Minecraft – Do beacons give buffs to friendly mobs

After some in-game testing, I have found that Snow Golems do not leave a trail on transparent blocks, including slabs. So you could replace your floor with upside-down slabs to keep the trails off. Adding a carpet will also keep the trails away.

Alternatively, you could light up the floor, but this may not be as practical, since that would require that you place torches on the floor every 2 or 3 blocks. So in this case, your best bet is to go with some transparent block for the floor.

Note: As of Minecraft 1.9 snapshot 15w46a Snow Golems will now no longer leave a trail of snow if the GameRule doMobGriefing is set to false.

Yes, it's called Iron Golem Farming and it's a widely used method of obtaining an infinite source of iron golems or iron ingots.

Quoting the Minecraft Wiki:

An Iron Golem farm is an easy way to acquire many iron ingots (and less useful poppies) with minimal effort after the initial setup. It is an artificial village (or several of them, spaced far enough apart to remain separate) in which golems are spawned and then either killed immediately, or moved to a holding cell (outside the village boundary, so that new ones can spawn in their place) for later killing.