How often should I have my chimney cleaned?

This a tougher question than it sounds. The simple answer is: The National Fire Protection Association Standard 211 says, "Chimneys, fireplaces, and vents shall be inspected at least once a year for soundness, freedom from deposits, and correct clearances. Cleaning, maintenance, and repairs shall be done if necessary." This is the national safety standard and is the correct way to approach the problem. It takes into account the fact that even if you don't use your chimney much, animals may build nests in the flue or there may be other types of deterioration that could make the chimney unsafe to use.
The Chimney Safety Institute of America recommends that open masonry fireplaces should be cleaned at 1/4" of sooty buildup, and sooner if there is any glaze present in the system. Factory-built fireplaces should be cleaned when any appreciable buildup occurs. This is considered to be enough fuel buildup to cause a chimney fire capable of damaging the chimney or spreading to the home.

My fireplace stinks, especially in the summer. What can I do?

The smell is due to creosote deposits in the chimney, a natural byproduct of woodburning. The odor is usually worse in the summer when the humidity is high and the air conditioner is turned on. A good cleaning will help but usually won't solve the problem completely. There are commercial chimney deodorants that work pretty well, and many people have good results with baking soda or even kitty litter set in the fireplace. The real problem is the air being drawn down the chimney, a symptom of overall pressure problems in the house. Some make-up air should be introduced somewhere else in the house. A tight sealing, top mounted damper will also reduce this air flow coming down the chimney.

When I build a fire in my upstairs fireplace, I get smoke from the basement fireplace.

This has become quite a common problem in modern air tight houses where weather-proofing has sealed up the usual air infiltration routes. The fireplace in use exhausts household air until a negative pressure situation exists. If the house is fairly tight, the simplest route for makeup air to enter the structure is often the unused fireplace chimney. As air is drawn down this unused flue, it picks up smoke that is exiting nearby from the fireplace in use and delivers the smoke to the living area. The best solution is to provide makeup air to the house so the negative pressure problem no longer exists, thus eliminating not only the smoke problem, but also the potential for carbon monoxide to be drawn back down the furnace chimney. A secondary solution is to install a top mount damper on the fireplace that is used the least.

I heat with gas. Should this chimney be checked too?

Without a doubt! Although gas is generally a clean burning fuel, the chimney can become non-functional from bird nests or other debris blocking the flue. Modern furnaces can also cause many problems with the average flues intended to vent the older generation of furnaces. We suggest you check the areas on gas and carbon monoxide for more information.

How common is it that chimney liners cannot be seen from inside the fireplace using only a flashlight? Is there some standard building requirement for the flue and the fireplace that you can't just look up from the fireplace and see the sky or chimney cap at the top of the chimney?

Flues are allowed to have up to 30 degree offsets. In most cases this will make a direct visual observation of the flue impossible. A video scan would be required to evaluate the flue condition.
The height of the chimney flue is not a factor. There is a big difference in what is observed between a visual inspection and a video inspection, even in short flues.