The lowest base movement speed in the game is 325. At this speed, Lich Bane grants 16.25 movement speed.
The movement speed stat only increases flat movement speed, and will not interact with other percentage increases.
Lich Bane's passive is considered an on-hit effect and, as such, is capable of being applied to skills such as Mystic ShotEzreal and ParrrleyGangplank, which effectively adds a 50% AP ratio to those abilities in addition to their natural AD/AP ratios.
The passive's 1.5 second cooldown will not begin after using an ability, but after hitting an enemy with the spellblade-empowered auto attack.
Similar to Trinity Force, Lich Bane synergizes well with cooldown reduction, enabling champions to proc the bonus damage as often as possible while minding the 1.5 second cooldown; the 10 second bonus damage buff does not stack. Using two skills in quick succession will not accrue two stacks of bonus damage, but one.
The effect by Lich Bane can be seen on the champion as a purple glow. For example, Kog'Maw will have a glowing mouth, Shaco will have glowing hands, and Master Yi will have a glowing sword, etc. The glow only appears after using an ability.
The same thing will occur with Sheen, Iceborn Gauntlet and Trinity Force. You can easily distinguish between Lich Bane buff and that of the other items by looking at the color.