Let's get some terminology straight. In equal temperament, octaves aren't merely perfect; they are "just" or "pure". "Just" and "pure" are synonyms while "perfect" has a different technical meaning in music. Nevertheless, "just", "pure", and "perfect" happen to be the same when applied to octaves. So your point #1 is correct.
When we speak of just/pure intervals, we are talking about the relationship between frequencies. Ascending octaves are always twice the frequency of the starting note. Descending octaves are always half the frequency.
When we speak of fifths, however, just/pure fifths and perfect fifths are not necessarily the same. A just/pure fifth is always higher than the starting frequency by a ratio of 3:2. So, if you start at 110 Hz, for example, the just/pure fifth is 165 Hz. 110 Hz is A2, so 165 Hz must be E3, right? In "just intonation", yes, it would be. But in equal temperament, it's about 164.813778 Hz. Either pitch would be the perfect 5th of A2, but only 165 Hz is the just/pure fifth.
Now that we have that basis, let's unpack your point #2.
However, when you divide each octave into 12 semi-tones, and tune it mechanically, there's going to be a problem with the fifths. Some of the fifths will be off.
That's correct because when you add up just/pure octaves and you separately add up just/pure fifths, they will never meet. (I asked for a proof of this on the mathematics stack exchange and got several great answers.) But it so happens that there are a few points where a stack of fifths and a stack of octaves come pretty close to one another. One such point is seven octaves and twelve fifths. Nevertheless, there is a slight difference. The twelve fifths are a bit higher. So we've come up with many systems to fudge the difference. These systems are called temperaments. Equal temperament is just one example. In this system, the same amount is taken away from all the fifths--hence it is called equal--so that the 12 fifths line up exactly with 7 octaves. Other temperaments handle this differently. Some fifths are (or are closer to) just/pure fifths while others are further apart. That is why, in other temperaments, each key has its own flavor whereas, in equal temperament, all the keys are identical when you compare them relatively.
I'm not sure: which fifths?
All of them, by the same amount. Instead of a fifth being 3:2 (a frequency times 1.5), a fifth is defined as the twelfth root of 2 to the seventh power. Since octaves are double, or a factor of 2, and an octave is comprised of 12 half-steps/semitones, we need to take the twelfth root of 2 to get an equally tempered semitone. Then, we multiply that by itself 7 times (because there are seven semitones in a fifth) to get the ratio of an equally tempered fifth. That gives us a ratio of about 1.498307 instead of 1.5. (You can do this on a calculator by raising 2 to the power of 7/12.)
To combat this problem (i.e. to get the octaves and fifths to agree with one another), a piano tuner lengthens the ... what, exactly? The distance between the notes in the lower half of the piano?
No, the fifths are all reduced as described above. You might be conflating equal temperament with stretch tuning, which is still based on equal temperament, but makes the lower notes slightly lower and the higher notes slightly higher.
If you're interested in more of the theory, I recommend reading this question. It goes into greater detail about the math, science, and the music theory. But you wanted a simpler explanation so that is what I've tried to give you.