Even if wind/solar/fuel cells/microturbines/diesel gensets or other distributed generation (DG) started producing 50% of the power, the utility generators would be rather easily able to to deal with this. Keep in mind that there's already something like a 50% (or bigger) change in demand every 12 hours, and they keep up with that just fine.
I'm gonna use "DG" instead of "wind" for the rest of this cause they're all really in the same boat, at least from the standpoint of the utilities and power consumers. Similarly, Google searches on DG & the various DG sources I listed above would likely do a better job of answering your questions rather than limiting yourself to wind.
Re: "However, if the availability of wind power comes when the demand isn't high, does this cause problem?" So long as the DG supply isn't greater than the total demand, the utilities will just back off until the supply equals demand again. It works this way mainly because those utilities are required to allow the alternative generators to sell back to the utility (not all utilities are required to do so -- here in Chicago, Com Ed does not allow export of power onto their system -- you can parallel, but not export).
If the DG capacity exceeds net demand, well, then some of the DG will have to go offline. I'm not aware of any location where small private DG capacity has outstripped demand yet, though, and I don't suspect we'll see that for a while.
DG generally tends to make things easier on the transmission lines, at least in terms of capacity. Putting a wind farm out in the middle of the California desert, and trying to run all of LA from it, though, would likely require additional transmission lines or upgrades to existing lines. Currently, there's not that much capacity in the existing lines to import power into California, which was part of the reason for their power crisis a couple years ago.
"1.) For example, could a significant amount of wind energy, occuring during low load periods, require the shutdown of a so-called 'must run' generating unit (i.e., nuclear, or coal units that don't cycle well, or units required to operate for transmission stability purposes)?"
Yup. And as I mentioned already, if you have 20GW of load, and 50GW of DG, then you'd have to shut down all utility output, and shut down 30GW of DG, too. But we're not there yet. And keep in mind, shut down just means take it off line, open the breaker. No, the utilities wouldn't like that, it would cost them not only shutdown expenses but a big loss on capital expenditures. But technically it's not that hard and not that different from the fluctuations in demand they experience every 12 hours. Now, if we ever get to the point where 30GW of DG needs to shut down, the big question would then be, OK, which one's gonna shut down? But most likely the answer will be that the DG guys won't be allowed to export, they'd only be allowed to generate what they used. Try to think of generation less as something being shoved onto the grid. Rather, think of it as a lot of capacity sitting there, ready to be used, and drawn off by customers as required. Unless the system gets overloaded, but that's a different story and one that's not applicable to your questions since you're dealing with overcapacity issues.
"2.) Similarly, could wind power during these periods cause the existing generating units to run less optimally (i.e., higher up their heat rate curves)?"
Well, most "centralized" utility generation is still distributed enough that they'd still be able to start up and shut down generators in such a manner as to keep most of them running at near peak-efficiency output levels.
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The biggest technical issues with DG typically have more to do with the interface between the DG and the utility. The utility is very concerned about the DG doing bad things to their system or exporting power at bad times. Protective relaying is often required to guard against this. Also, on smaller systems (such as residential solar systems in California), there's new IEEE & UL standarda set up (I don't remember the numbers off hand, but could dig them up if you can't find them on the net) that permit you to get all the protection you need built into the inverter. Just use the listed inverter, wire it up in accordance with the directions, and you're set.
On bigger systems, relay packages are typically used to guard against some of the following conditions:
1. Electrocution. The utility lineman might open his switch, and think his line has been isolated and it's safe to work on. But the DG keeps it energized. Bad.
2. Islanding. If the utility goes out (and I suspect you have first hand experience with utility outages regardless of where you live), the DG will try to support the surrounding neighborhood. On larger DG systems, it might even be partially successful in this attempt. And this might be a problem either for the DG generator or for the utility. So protection against islanding is required.
3. Ground Faults. If the utility has a ground fault, then the DG will also try to feed into that fault.
4. Synchronization. If any two generators anywhere try to parallel and they are not synchronized and in-phase, then massive overcurrents result. Bad.
Balancing of voltages and reactive power flow can also be issues.
In general, DG has a lot of advantages to offer, and I think we'll see a strong steady rise in DG for a while. The utilities also have a lot of advantages, particularly with fossil fuels and nuclear fuels -- not the least of which is economy of scale. I think we'll find that the utilities and DG can peacefully exist for a long long time.