I am wondering if I may be causing harm to my new Uno when I power it up or when I either yank out its power from my Mac laptop or pull a wire or two from the two-inch breadboard.
To power up, should I follow a certain order of hooking up the components?
As others have noted, you should have your circuit set up and ready to go (and double/triple checked!) before hooking up power. In the case of certain components (CMOS devices, for instance), you will want to wear a grounding strap, connected to a suitable ground, while working with the components. In this situation, you -may- also want the ground of your breadboard connected properly to where the ground of the component connects (this is really a paranoid practice, I admit - but if your component is an expensive instrumentation amplifier or something similar, it might be well worth it while prototyping).
For example, I connect one end of the USB cable to the laptop; then I hook its other end to the Uno.
This order isn't really too important - with one exception: USB ports are notoriously fragile with repeated connect/disconnect cycles. So you need to ask yourself which you care more about: The port on the UNO, or the port on your computer? If you care about both (why not?), then keep a cable connected long-term to the UNO, and plug/unplug it from a cheap hub connected to the computer. That way, the port to "fall apart" over time will likely be the one on the hub.
Then I insert the + wire from the breadboard to a digital pin __, and then the - wire from the breadboard to ground. Is this electrically proper, or does the sequence even matter?
Provided that there isn't any power to the Arduino, it likely won't matter (the exception could possibly be the paranoid wiring method for static-sensitive devices mentioned above).
Is there any difference at all if I disconnect power by pulling the USB cable from the laptop end or from the Uno end?
In most real-world terms, no. The only thing to be aware of is the fragility issue noted above.
Does it harm the Uno if I disrupt its circuitry while it is powered on, like pulling wires from it, adding wires to it, adding or swapping LEDs and resistors to the breadboard?
As others have noted, you should get into the habit now of -not- doing this; you might get lucky for a long time, but building/altering/configuring a circuit while things are powered on is a recipe for disaster. Right now, at most you are limited to maybe a small pop, and a puff of smoke (if even that) coming from some part or the Arduino, should you make the wrong connection. Not much monetary or physical damage. However, if you kept up that habit, and moved on to circuits carrying larger current and/or voltage potentials, everything from electrical shock, to fire, to even injury and/or death could easily occur.
Back in the day of television repair - there was more than one repairman who either was seriously injured or killed while working on a set while it was powered up; sometimes the injury or fatality had nothing to do directly with electrocution - at times, it was more from the sudden jerk of the muscles from the shock that caused the person to be "flung back", cracking their skull or other body part on the workbench, floor, or elsewhere.
Which leads to another rule you should be aware of, the so-called "one-hand" rule.
What this means is that when working on an electronic circuit, always probe and test the circuit (especially if it is live - which you should strive to avoid, but sometimes it isn't possible) using one hand. Keep the other hand away from the circuit under test (and in your pocket if possible). This rule is especially true if you are working on anything with high voltage or high currents (especially in high current situations). The reason for this is to prevent a path for electrical conduction that runs across your heart (thus potentially inducing a defibrillation event should electrical shock occur). It also can prevent damage to components, because it is fairly easy to have a short-circuit occur when using two probes at the same time versus a single probe. Ground or connect the second probe to where it will do the most good for your testing, and use a single probe in one hand only.
It is best practice, though - when you can - to not probe a live circuit period (rather, hook up your probes before applying power).
Finally - one last thing: Never assume a circuit is dead just because you think there isn't power going to it. This is much more important on higher-power circuits where you don't know the layout or can't easily see the layout (perfect example: house wiring). I once lived in a house where I had to install a new bathroom lighting fixture. To do this, I was going to simply move the existing fixture (an overhead light that was poorly placed) to a new fixture over the bathroom mirror. This meant that I had to do an attic crawl in order to move the wiring properly from the ceiling, to the wall (I also had to bore a hole in the top-plate of the wall, too). Now, normally this meant turning off the breaker to the bathroom, and turning off the switch in the bathroom. Well, I still didn't trust it for some reason. I hooked up my neon tester to a portion of the light circuit while I was up there, before I connected anything - and I still had power! Turns out that someone before me had hooked up two circuits to the same lighting circuit (this is a big no-no in electrical code). I ended up having to find that breaker as well, shutting it off, then figuring out the path to remove that circuit from the lighting circuit being used for the fixture - just to install a simple bathroom light. Had I not been paranoid and double checked the power...