Short version:
Evasion?
Mostly no
Critical Strike?
Partially
Bash?
Partially
Increased attack speed?
Yes
Flat ms boni?
No
Flat hp regen?
Yes
Long version:
Evasion:
There are two different ways of evading an attack by ability (the only other real way of evading an attack is the higher ground mechanic. Exceeding motion buffer range, hiding a unit, etc should not count as real evasion):
1. Normal evasion (Evasion, Drunken Brawler, Bash, Critical Strike), which is on the target
2. Blind (Curse, Cloud, Silence, Drunken Haze), which is on the attacking unit
Normal evasion does not stack, the highest value of the target will be used.
Blind does not stack, the highest value of the attacking unit will be used.
Normal evasion and blind (and higher ground) do stack in the normal diminishing returns sense:
1 - (1 - evasion chance) * (1 - blind chance) = overall chance to evade the attack
Critical Strike and Bash (Drunken Brawler, Bash, Critical Strike):
Quite complicated, you don't want the complete version.
This gives a good overview though (you have to scroll down a bit). If you want you can read on with
PRD, which is used by crits. This is still not the final truth but most of it is accurate.
Increased attack speed (Gloves of Haste, ...):
IAS add up and can range from -0.8 to +4, starting with a value of 0 for no bonuses. For a normal unit the following formula can be applied (with BAT being base attack time):
Attacks per second = (1 + IAS)/BAT
Flat movementspeed boni (Boots of Speed, ...):
Does not stack, independent from the gameplay constant "MoveSpeedBonusesStack". The constant has no known use.
Flat hp regeneration (Ring of Regeneration, ...):
Yes.