What separation and information does ATC provide in different airspace classes?
Different countries have different meanings to the classification of Airspace. I am only looking at the controlled airspace for ICAO.
Controlled airspace is an airspace that an aircraft would need a clearance to enter, this doesn't mean all aircraft need a clearance to enter just some aircraft.
Classes of Airspace
Class A is for IFR flights only, all aircraft are separated from all aircraft and are known to Air Traffic Control (ATC).
Class B is like Class A but with VFR. IFR and VFR are cleared to enter, all aircraft are separated from all aircraft and are known to ATC.
Class C is where the major confusion on separation usually begins. IFR and VFR are cleared to enter, all flights are provided an ATC service. IFR flights are separated from IFR and VFR flights, but VFR flights are only separated from IFR flights. VFR aircraft are only passed traffic on other VFR aircraft. All aircraft are known to ATC. I will go into what they do when they get in conflict at the end.
Class D both IFR and VFR are cleared to enter, all aircraft are provided an ATC service. IFR aircraft are separated from IFR and passed Traffic on VFR flight. VFR flights are only passed traffic on other aircraft. All aircraft are known to ATC.
Class E IFR are cleared to enter and VFR are permitted to enter, only IFR aircraft are provided with an ATC service. IFR aircraft are separated from IFR aircraft and passed traffic on known VFR aircraft. VFR aircraft do not need a clearance and are not usually known to the Air Traffic Controller.
Class F No one is given a clearance. IFR aircraft are given traffic on IFR Aircraft and known VFR. But all pilots are responsible for their own separation.
Class G No one is given a clearance. Traffic is only passed where it is requested, and only on known traffic. Only those requesting traffic or are seen on a radar are known to ATC.
Answers to Specific Questions

I fly a rotor-wing at a class D airport. Under VFR, we should only receive &quot;traffic information&quot;, correct?

Yes, that is correct.

But in reality, we still have to request ATC's permission to take off etc. and they can instruct us to hold, give way and so on. Doesn't that mean that we are in fact being &quot;separated&quot; from other traffic?

When talking about Class D airspace: you are only separated from other aircraft when on the runway; when you are in the air you are not being separated. You instead will be passed Traffic Information on other flights. You however are still subject to the conditions of your clearance (Route, Level, Holding Instructions, etc.).

And theoretically, if IFR traffic is controlled by ATC in class D but VFR is not, and IFR is not separated from VFR, how should conflicting traffic avoid each other? If an IFR flight sees a VFR one in its way, who should do what? (Practically, the pilots will obviously maneuver to avoid each other, but what does the theory say?)
Finally, since IFR flights only get &quot;traffic information&quot; about VFR flights but not &quot;separation&quot;, what happens if a VFR flight gets in front of an IFR one on final approach to land? Won't ATC tell the IFR flight to go around, or will they do nothing because they don't provide &quot;separation&quot;?

If one or both aircraft are under a clearance the cleared aircraft will request an amended clearance to avoid the traffic:
ABC request 10NM right of track due traffic

or
ABC request descent 4500 due traffic

If there is not enough time for the amended clearance the Pilot in Command (PIC) can just break their clearance to avoid the traffic; either passing the aircraft on the right or changing levels. The PIC in some countries will be considered to have broken their clearance but it is usually only a slap on the wrist for letting it get that bad that you couldn't request the clearance change.
Reality can sometimes be a very different scenario. Some controller unofficially will separate VFR from IFR, but no one knows. Don't trust that this will happen.
Also controllers may give suggested amended clearances like:
ABC traffic XFV PC12 crossing track left to right at time 23,
6500 is available advise.

or
ABC traffic unknown VFR 25NM oposite direction 500 ft below unverified,
higher available advise

If the aircraft is not subject to a clearance then they can just move to avoid the traffic under the Rules of the Air.
In uncontrolled airspace it really is see and avoid. Because the Air traffic Controller may not have all the aircraft that are traffic for you. This also includes Class E airspace where the controller doesn't know about all the VFR aircraft.
Duty of care means that the controller may intervene if they think the pilots aren't separating themselves.
I work with Class D and C Towers with lots of VFR traffic. Because the traffic are aware of each other they usually request amended clearances well before the traffic is a problem.