My bona fides (so to speak): Worked at Yucca Mountain, have friends in the civilian nuclear industry (plant operators, nuclear engineers, experts in spent nuclear fuel). Been following the industry for 15 years. Not an expert (couldn't jump a feedwater valve unless you told me how to do it), but I consider myself as having above average knowledge - enough so to be able to understand what experts are saying when they go techno-babble.
I'll answer your questions and then provide detailed explanations below.
Short answer: Depends.
EMP - Yes, meltdown.
CME - Probably meltdown, depends on damage to plant equipment.
Standard Outside Electrical Blackout Long term - Maybe, depending on availability of diesel fuel and ability for NPP to provide its own power.
Standard Outside Electrical Blackout Short term - No, provided diesel generators are functional.
A total loss of electrical power, called a plant blackout, is one of the most serious threats that a NPP can face. Probably the only greater threat is an unexpected catastrophic failure of primary coolant machinery or piping or significant major component malfunction or failure.
You have to remember that NPPs have a series of redundant features. For your typical Pressurized Water Reactor (PWR), you have outside electrical power, emergency diesel generators, and finally battery power. I understand, through conversations with plant operators and nuclear experts, that if there is enough time available, the switchgears between the electric yard and turbine can be connected, allowing a reactor to essentially provide it's own power.
Further, you have to remember that when it comes to a meltdown, we are concerned about two different systems. The first is the power generation system (the reactor), the second is the spent fuel pool (used fuel storage). The two systems are fundamentally different, with different containment systems, redundancies, and strategies for dealing with a plant blackout.
Let's look at the power generation system first.
First, I need to point out that there are typically three levels of building containment in a power generation system. First, you have the Reactor Pressure Vessel. This is the actual vessel that houses the components where the nuclear reaction takes place. Second, you have a steel containment building. This surrounds the Reactor Pressure Vessel, Steam Generators, and any stuff that come into contact with nuclear materials, including coolant. Third, outside of the steel containment, you have a concrete containment. The steel and concrete containment structures are referred to as Primary Containment.
What a plant blackout does is lead to a Loss of Cooling Accident (LOCA). Simply put, coolant is needed for both the reactor and for the spent fuel pools. Reactors create a tremendous amount of heat - that's what they're designed to do. Essentially, a nuclear reactor is a way to boil water, which turns a turbine, which turns a generator, which produces power. That's it.
Without coolant, the reactor overheats, and the fuel assemblies and reactor guts melt. This creates something called corium - think of corium as radioactive lava. It's a molten mix of nuclear fuel, metal (reactor guts), and some other things that melt when the reactor gets too hot. Eventually, the corium pools at the bottom of the Reactor Pressure Vessel (RPV) (which is the steel can that contains the reactor - like how a can contains your soda), and it may melt through (Chernobyl), or it may not (Three Mile Island).
If the corium does melt through the RPV, it usually drops into a containment pit (also called a Core Catch), which is designed to prevent the corium from getting outside of primary containment. I can get into the details if you want to, but it's really not relevant to this post.
The other danger is that gasses are created during the process of a nuclear meltdown. If the RPV or other component of the primary coolant loop is compromised, then these gasses can vent into Primary Containment. While PC is designed against an explosion blowing it out, there is a chance that PC could be compromised.
If outside power is off, then the plant can run off of diesel generators. If the diesel generators are unavailable, then batteries provide a limited amount of power. When the batteries run out, then passive cooling systems attempt to keep the damage to a minimum.
I've omitted a whole bunch of things in the above explanation. Emergency Core Cooling Systems, Passive Safety, Low Pressure Coolant Injection Systems, High Pressure Coolant Injection Systems, Depressurization Systems, Core Sprays, and Isolated Cooling Systems. I can get into that if you want - but it pertains more to reactor design than it does to this situation.
That's the power generation side in a nutshell.
Onto the spent fuel pools.
The spent fuel pools are essentially gigantic swimming pools (well, they look like swimming pools - you wouldn't want to swim in them unless you had a death wish) where used fuel assemblies are stored. They are stored there for two reasons. First, used fuel assemblies are incredibly hot and continue to generate heat even after they are removed from the reactor. This is referred to as Decay Heat. Second, you need a lot of shielding between you and the fuel assemblies - water acts as a shielding for this purpose. Thus the water serves two purposes - it cools the fuel and it's a radioactive shield.
In a plant blackout, what happens is that the water that is lost due to evaporation is not replaced, and the water is no longer circulated mechanically (there is some passive water circulation, but I won't get into that). The spent fuel heats up the water (generally - more on the exceptions in a moment), eventually causing it to boil off. The water level drops, the fuel becomes exposed, and it eventually either catches fire and/or melts down. In a very extreme case, you may have an uncontrolled nuclear reaction occur (note - this is not similar to a nuclear bomb).
Some, but not most, spent fuel pools are designed to radiate enough heat that you really don't need to put water in them. Once it goes in, that's enough, and there are passive systems to replenish the water supply. Most, to my knowledge, are not, because we were supposed to have a place to dump this fuel by now (Yucca Mountain), and we don't, so we've been forcing NPPs to keep spent fuel on site.
The spent fuel takes anywhere from 2 to 7 years to cool adequately. For safety reasons, you'll never hear a nuclear expert say anything less than 5 years, but I've been told that they could do it in 2 if they really had to. After the fuel has spent the appropriate amount of time in the pool, they move it to a dry casket and store it outside.
Most spent fuel pools - but not all - are stored in containment buildings similar to Primary Containment found around reactors.
Hope this makes sense.
Let me know if you have any questions.