The slowed condition reduces your speed to two. However, there are some powers that allow you to move a specific number of squares regardless of your speed, or move an amount of squares based on a non-speed character stat. Would these powers ignore the effects of the slowed condition?
Yes. Powers that have specific speeds described in the power ignore the slowed condition.
The text of slowed reads:
When a creature is slowed, its speed becomes 2 if it was higher than that. This speed applies to all of the creature’s movement modes (walking, flying, and so on), but it does not apply to forced movement against it, teleportation, or any other movement that doesn’t use the creature’s speed. The creature also cannot benefit from bonuses to speed, although it can take actions, such as the run action, that allow it to move farther than its speed.
Thus, when powers say "Shift 3 squares" the power does not use a specific movement mode speed, and therefore is not impacted by the condition.
In general, yes. Speed is a specific thing and if the slowed condition were meant to apply to all forms of movement, rather than changing speed, it would say that.
That said, as a DM I would evaluate individual "move N squares" or "move based on Z factor" effects to see if I thought that what's generating them would also be affected by the slowed condition. If it were something that was fundamentally still based on the character's physical capabilities, I would probably cut it down to 2 squares or impose a partial effect.
This is, of course, only relevant if one is trying to be the sort of DM who assigns primary responsibility for the integrity of the setting to himself or herself, rather than the rules.