If staggered you only get one standard or move action, can you still 5 ft step?
Yep sure can, 5 foot step is essentially a free action. Anything more than a 5 foot step though and your looking at continued movement which takes your move action.
Yup.
As a 5-foot step is not considered an action (according to the Action table in the Combat section), you can still take a standard and a 5-foot step.
Although, remember the rules about making a movement and 5-foot steps.