How to climb stairs

• The attack is only triggered by enemy movement. Moving on your turn will not trigger Bladestorm (unless your move causes an enemy to move, such as a burrowing Chryssalid).
• No, only the movement will trigger Bladestorm. I've had an Andremedon move past my soldier (triggering Bladestorm), then in a separate action, make a melee attack. The attack was not responded to by my Ranger.
• They happen immediately, usually before the enemy gets to take their action. This is very useful against, e.g. Stun Lancers, which will move into attack range but die before they can actually make an attack. Some enemies, such as Faceless, have a melee range larger than the Ranger's, and don't actually have to move close enough to trigger Bladestorm to attack. (It seems there may have been a recent patch affecting this; it's possible for Bladestorm not to trigger until after the enemy gets to attack.)

Alright, I dug into the code some more and updated my answer.

Here are the number of kills needed for each level, taken from DefaultGameData_XPData.ini:

            Rookie  Veteran  Commander  Legend
Squaddie   |   1  |    1    |    1    |    1
Corporal   |   5  |    6    |    6    |    8
Sergeant   |  12  |   12    |   12    |   18
Lieutenant |  24  |   24    |   24    |   40
Captain    |  38  |   38    |   42    |   70
Major      |  57  |   57    |   60    |  110
Colonel    |  81  |   81    |   85    |  175

However, there's more to it than just straight-up kills.

Every time a unit gets a kill, all the other units on that mission get partial-kills. The game calls these "KillAssists", but that's a misleading name because the unit earns a KillAssist whether or not they did any damage.

How much a "KillAssist" is worth varies by class
(taken from DefaultClassData.ini, variable name KillAssistsPerKill)

Specialist:    1/3rd kill
Psi Operative: 1/3rd kill
Rookie:        1/4th kill
Ranger:        1/4th kill
Sharpshooter:  1/5th kill
Grenadier:     1/5th kill

Additionally, Psi Operatives gets a kill bonus called a "PsiCredit", worth 1/3rd kill, every time they are hit by Feedback.

Finally, if a unit started at any rank above 'Rookie', they'll have additional hidden bonus kills equal to the number necessary for that rank.

Assuming you obtained a unit at 'Rookie', you can see how many kills they have in the armory...

but without a save-file viewer (which doesn't exist AFAIK) you cannot view their KillAssists. The best we can do is guesstimate from the number of missions.

In my screenshot above, if we assume she's encountered, say, 8 enemies per mission on average, then over 10 missions she's encountered 10*8=80 enemies. That gives her 80-29=51 KillAssists, which counts for 51/4=12 kills, bringing her approximate kill-total to 29+12=41 kills. So she needs around 16 kills or 64 KillAssists to level up to Major.

Edit: There is now a mod that shows KillAssists on the soldier info screen: Accurate Soldier XP

Some other interesting things I discovered while code-diving:

• Except for rookies, units only gain levels after actually getting a kill, so you can't have a Colonel with 0 kills.
• Kills are still kept track of after getting a promotion mid-mission, so people who claim "When a unit gets a promotion further kills are wasted" are incorrect. However, it is true that a unit can only gain one level per mission, even if they pass the threshold for two levels. In that case, they'll get promoted immediately on their first kill in the next mission.
• As of the latest patch, Wet Works (25% XP boost) is no longer retroactive. KillAssists and PsiCredits are not boosted by Wet Works.
• Despite popular claims to the contrary, there is no XP in this game, it was disabled before the game's release (bUseFullXpSystem=false). Though, I suppose you could consider KillAssists and PsiCredits a form of XP...
• I've created a mod that gives a small number of kills to idle units (for technical reasons I can't add KillAssists). There's an option to allow them to level up outside of missions as well.