Minecraft – How much damage will the Snowball do

Lightweight version
It seems I overcomplicated things a bit on the first attempt, this version should be better and more compact, at least for smaller numbers (first example produces 4 pulses). It's also possible to connect the "start" for another pulse straight from the second torch).

For this setup to work with a high pulse count you will need to put repeaters every 15 blocks in the redstone line, and for each of those repeaters there will need to be extra ticks between all stone blocks. This exact setup will work for your 11 pulses (last 6 blocks of 11 shown):

Heavy version (no limit)
Basically an outer "zig-zag" circuit that will send a new pulse straight through the middle line each time it passes a stone block. I tested it with up to 11 pulses, and you can just extend it as much as needed. Obviously it gets quite clumsy with a large number of pulses.

No. In vanilla Minecraft on a vanilla server, snowballs and other throwables do not hit players.

It's simply occam's razor; if snowballs didn't hit the player in alpha 1.0.5, and don't hit the player in release 1.8.4 (and also all versions after release 1.4.4, as confirmed in MC-3179), and there is no indication in any changelog that the behavior changed, let alone changed twice in between those versions, it stands to reason that such a change did not occur.

There are some server plug-ins that change this behavior, though. Snowball Damage is a Bukkit plugin that does precisely what it says on the tin. Similar plugins with similar functionalities exist. This is likely where the confusion comes. Many, many servers use Bukkit (in 2012, 80% of servers were running Bukkit. More current statistics are unavailable), so many users may have been exposed to such plugins, and are remembering behavior that does not exist in vanilla.