Meanwhile, 5-ft step is a miscellaneous action. Is it allowed, by RAW, to make a standard action and a 5-ft step (provided that half speed is greater than 5ft) when slowed?
Yes in D&D 3.5e
In D&D 3.5e, a 5-foot step is not a miscellaneous action — it is not action type at all. What it is, is an option you may exercise as a consequence of not moving. The trigger to be permitted a 5-foot step is:
If you move no actual distance in a round
If that is true, you can make a 5-foot step (barring other restrictions that prevent exercising that option once permitted, of course).
Maybe in Pathfinder.
In Pathfinder, the 5-foot step is explicitly a Miscellaneous Action. Since staggered only permits free, swift, and immediate, miscellaneous actions are seemingly barred.
But when you take a step back an analyse the nature of Miscellaneous Actions, it turns out that it may not actually be an action type at all:
The following actions take a variable amount of time to accomplish or otherwise work differently than other actions.
You see this further in the other two categories (feat and skill use), which say that they take whatever actions types they say they do, or maybe none at all.
What this indicates is that "Miscellaneous Actions" isn't an action type so much as a bucket for other stuff that doesn't properly fit into the action types, the action economy, or both.
So: maybe. To interpret it as a "yes", you have to look at the Miscellaneous Actions section and conclude that a 5-foot step is not a special action type, and then you get the same conclusion as for D&D 3.5e, above.
To interpret it as a "no", you have to interpret the title of the section "Miscellaneous Actions" as creating a new action type; but this has its own issues with complicating the understanding of skill and feat use:
- is a skill/feat that says it takes a standard action simultaneously a miscellaneous as well as a standard action?
- is a skill/feat that says it takes no action simultaneously a miscellaneous action as well? is it an action or not, then?
Since the "no" results in a multiplication of mechanical entities within the rules, and they potentially lead to self-contradiction, both Occam's Razor and a hint of reductio ad absurdum leads me to favour saying "yes," a 5-foot step can be made when staggered, just like in D&D 3.5e.