The essential difference is in whether the action is starting or ongoing (do) or is finished (have done).
Can I have the newspaper when you've read it?
Only "have read" makes sense here, since you probably mean you want the newspaper after the other person finishes reading it. If it were "read", it would mean you want it while they simultaneously read it.
Can you let me know as soon as you decide? / as soon as you've decided?
Either one is acceptable, because "decide" is not continuous, but instantaneous — starting and stopping are the same thing. I would say they're about equally likely as well, though the present perfect somehow strikes me as more polite / formal.
I think the commute will be better when they they finish / when they've finished the new road.
For the same reason as "decide", either one is acceptable. But if you were to use "build" instead of "finish", then you would have to use the present perfect, since the present would mean that they commute will improve simultaneously with the road being built, which is unlikely.
Would you like a cup of tea before you leave / go?
Here "before you've left" or "before you've gone" would not work because it opens the possibility that they have a cup of tea between starting to leave and finishing leaving, which makes very little sense. Indeed, if you understand "leave" as an instantaneous verb like "decide", then that interval of time implied by the present perfect doesn't exist.
I'm not sure if this logic would hold up in every case, but it seems like a good fit here.