Exactly when do I level up in sneak?

Aside from trainers, quests and books, sneak is levelled in two ways.

First, sneaking close to someone will increase your sneak skill if they cannot see you. You can earn experience quite easily this way by sneaking into a wall near someone who is unlikely to turn around or detect you. Sleeping people are extremely good choices for this.

Second, sneak attacking people earns you sneak xp. You can get a lot of xp farming plot-critical characters such as the Greybeards this way.

"[H]ow close do I have to be to someone?"
~45 feet.

"Do I level up faster if I'm closer?"
No, I don't think so.

• 2.5 base XP for becoming hidden within ~45 feet.
• 0.625 base XP per second hidden within ~45 feet.

http://www.uesp.net/wiki/Skyrim:Leveling#Skill_XP

"Does being partially noticed help or hurt?"
Help, I guess, because you gain Sneak XP while "being partially noticed", apparently even without having to move at that point, unlike if no one even suspects your presence.

Aside from sneaking while "being partially noticed", there are multiple ways to gain Sneak XP:

• moving in a sneaking state within earshot of a non-friendly but not being (fully) detected,
• sneak attacking,
• getting training from a Sneak trainer,
• reading certain skill books,
• certain quest rewards, and
• the "advSkill" console command
• (, and also some anomaly at the beginning of the game).

Using a skill grants skill XP according to

Skill Use Mult * (base XP * skill specific multipliers) + Skill Use Offset

Sneak Skill Use Mult = 11.25
Sneak Skill Use Offset = 0
Sneak Skill Improve Mult = 0.5
Sneak Skill Improve Offset = 120
∴ 11.25 * (base XP * skill specific multipliers)

Sneak "Sources of XP/Notes":

• 2.5 base XP for becoming hidden within ~45 feet.

• 0.625 base XP per second hidden within ~45 feet.
• 30 base XP for a melee sneak attack.
• 2.5 base XP for a ranged sneak attack.
• Additional multipliers may apply.

http://www.uesp.net/wiki/Skyrim:Leveling#Skill_XP

Range, by my own testing:
Enemy unit: X -396.68, Y 4701.32, Z 471.82
Me within edge of range: X 967.24, Y 4678.92, Z 400.65
Considering how far one unit does not move me, I'd guess that the units are in inches. Considering how flat the ground seems to be and the difference between my z-pos near the enemy (384.5) and at the edge, there seems to be a difference between the enemy model's (Imperial Soldier) z-pos and mine (Imperial female with weapons drawn). Just calculating the X and Y axes, their range of perception is sqrt(1860779.5264) or 1364.10 units. If the units are inches, that's 113.675 feet.