Can you lose the legend rank in Hearthstone

The number of cards and the rarity is fixed, the actual cards you get is random.

Rank 20-16: 1 golden common
Rank 15-11: 1 golden common, 1 golden rare
Rank 10-6:  2 golden common, 1 golden rare
Rank 5-1:   2 golden common, 1 golden epic
legend:     3 golden common, 1 golden epic

Additionally you get between 5-25 dust.

The important thresholds are rank 15 and rank 5. On rank 15 you get a golden rare (100 dust), and on rank 5 you get a golden epic (400 dust).

Source

Yes, other players see your skin instead of the default hero for your class!

They also hear the different voiceovers and emotes of course, that's what skins are for.

I can't really confirm your experience to be honest, I personally see them a lot when playing ladder. When the skins came out it was actually very rare to see a regular hero instead of the alternate one. ^^

I have to mention that I usually play on the higher ranks, so my opponents are playing for a longer time as well, increasing the chance that someone there bought a skin. In the lower ranks there are a lot of new players which probably don't want to pay money for cosmetic stuff when they can spend the money on card packs. However, this is just a wild guess but I think it goes to the right direction.