When are pot lids useful?

Covering a pan slows down evaporation of the water. As evaporating water takes a lot of heat, covering a pan will make the contents boil earlier.

If the recipe tells you to cover the pan, it's to make sure there's enough moisture for the full duration of the cooking; without the lid, you might end up with a dry (and usually burned) mess.

When cooking rice, the proteins and starch dissolving in the water will cause the bubbles that form on boiling to be more resistant, so they form a foam that lifts up the lid and escapes. Usually after the first foam is destroyed, you'll have no more problems (but do keep the fire low). And for at least one method of cooking rice, you want to keep the pot well covered, as the amount of water you add is just enough to fully cook the rice (the water is absorbed by the rice and makes the starch grains swell, without water the rice would stay very hard).

It all boils down to how fast you want the water to evaporate.

I generally agree with remco's answer. Lids will allow things to heat up faster, and they are useful for retaining moisture. Many recipes that advise you to use a cover are depending on most of the moisture to stay in the pan during cooking (and, particularly with things like rice, the proportion of liquid may be important).

That said, there are times when lids are also less important or even should NOT be used. For quick-cooking dishes, evaporation is often less of an issue. And sometimes you want to boil off excess liquid. If you don't mind the liquid/sauce/etc. being reduced a bit, you may be able safely ignore recipe instructions to cover.

Keeping steam in a pan will also lower the effective cooking temperature (and keep it around the boiling point of water), which may be less desirable in many situations. For example, if you're sauteing or trying to brown things in a pan, a lid is counterproductive. If you're going to put a pot in the oven to cook for a long time and want the surface to brown a bit and develop flavor (sometimes useful for stews and such in the oven), you might let the pan uncovered for some part of the cooking so the surface of the food can reach a higher temperature. (However, keep in mind that over a long time, evaporation will also carry some flavor away with it.)

A few practical answers to points raised in the question:

• Rice (and many other grains) can actually be boiled in excess water -- like pasta -- and then drained. Some nutrients may be lost (though there's not a lot of nutrients in white rice anyway), but you don't need a lid to do it if you use that method. It can also be helpful for people who have difficulty burning rice or not cooking it properly. It doesn't work well with short-grained "sticky" rice, but it's common to cook this way in many cultures with medium or long-grain rice. And just like cooking pasta, you can easily take some out to taste and make sure the texture/doneness is to your liking.

• Generally speaking, the need for "tight-fitting lids" is rare. Assuming the "lip" on your pan is level (and it should be unless your pan is damaged or warped), you can use lots of flat things to cover a pan or pot. For years, I had a single "universal" lid I used on many pots. A few years ago, I bought a cheap set of three glass lids of varying size which I use most frequently when cooking. (The glass allows me to see the contents, which is helpful sometimes.) They don't fit any of my pots precisely, but they're perfectly adequate for almost all stovetop cooking. I own many somewhat pricey pots (mostly copper) which I don't have custom lids for. Having a matching lid for every pot and pan in your kitchen is generally unnecessary and can take up a lot of excess storage space.

• One tip if you are going to buy "generic" lids: find lids that are simply flat and have a little weight to them (to avoid "rattling" and letting steam out). It's generally better just to have a flat disk, rather than to try to use an ill-fitting lid with curved edges or something designed for a different pot. The key is just having a disk-shaped lid at least as big as the pot. The only time you'll get significant moisture loss is if you try to use an ill-fitted lid designed for a smaller pot.
• All of that said, it's often good to have at least one or two pots with actual lids to fit them, particularly if you're going to be cooking stews for 3 hours or something. A well-fitted lid can have subtle design features to prevent excess evaporation, and that can be helpful for long simmering. But I also frequently cook brown rice for 45 minutes using a generic glass disk lid that doesn't fit my saucier, and it all works fine with seemingly little problem in losing water.
• As for rice boiling over, make sure you're using a big enough pot. And watch somewhat carefully until the water is boiling with rice in it. Then make sure the heat is very low. (I usually just wait until the water returns to a boil after adding rice before putting a lid on anyway, which probably is common to many directions for cooking rice.) On any stove, it usually only takes me once or twice to figure out the appropriate heat level to keep rice simmering for a given batch size. If you keep the heat at a low level, it shouldn't boil over. (And again, if you have trouble with this, try cooking in excess water uncovered and then drain when it's done.)