How do countries like Denmark, which apparently consider using equipment with a three-prong plug in a two-prong socket, treat the difference between IEC protection classes? In many countries where grounded outlets are common, equipment either needs a ground connection (class I), or has to apply to more stringent double insulation standards to reduce the chance that any single fault will bring the user into contact with dangerous voltages. Double insulation is getting increasingly popular because preventing contact with mains wires is considered safer than blowing the fuse in case of contact.
I can not answer for Denmark nor Norway, but it seems the situation there is similar to Sweden. In Swedish installations an area (usually a room) has either grounded outlets or ungrounded outlets, but according to code is not allowed to have both. New installations are required to use grounded outlets and RCD/GFCI.
Class II appliances use plugs that fit in both types of outlets and are allowed in both grounded and ungrounded areas.
Class 0 appliances have plugs that only fit in ungrounded outlets and are only allowed in ungrounded areas. The thinking is that there are two safety barriers: one is the internal single isolation barrier of the appliance, and the other one is the relative inaccessibility of ground in the environment, preventing a closed current path. Such appliances may no longer be legally sold.
Class I appliances may be used in both kinds of areas. In an area with grounded outlets, the protective earthing is relied upon as the second safety barrier, while two-layer safety is provided in exactly the same way as for Class 0 appliances if used in an area with ungrounded outlets.
So basically you are allowed to use an appliance in an area if its plug fits. Of course there might be other restrictions, such as appliances designed for indoor use only.
Can you just design equipment to the lower class I creepage/clearance standards without providing a ground connection in countries like Denmark?
That would be a Class 0 appliance, and selling them are no longer allowed in Sweden. Among other things, it would not be possible (without illegaly modifying the plug or similar) to use such equipment in a modern installation which has only grounded outlets. A Class I appliance needs to have the grounded cord and plug, regardless of where the user is going to plug it in.
Do manufacturers like Agilent or Rigol test their equipment for safety and EMC/EMI without ground connection before marketing it in Denmark? Are there building code requirements about a minimum impedance to ground, just like the maximum earth ground resistance required in other countries?
Sweden allows equipment designed to the Harmonized European Class I and Class II standards to be sold. There is no additional requirement. The safety of Class I equipment in ungrounded areas is no worse than Class 0 items in the same areas. As for ground impedance, equipment is governed by the same standards. Electrical installations have requirements that the fuse/breaker must trip within a certain time if there is a phase/ground short at the load, setting a limit to how high the grounding impedance may be.
Anyways, I think that for an electronics lab grounding is preferred, even if not having it would at least be relatively safe especially as an RCD is installed. Just as Mike wrote in the second post of this thread, there is a risk of damaging sensitive electronic devices when making connections. The ground connection is also preferable to have to be able to effectively discharge static electricity for ESD protection reasons.
Having your local "ground" floating (usually about 110 V AC due to those mains filters and other capacitive coupling in equipment) also tends to couple hum into high-impedance nodes of circuits being measured/tested. So there is also a signal quality problem.