(1) A kanji's meaning is the same regardless of reading.
Not always true, and in many cases it's hard to talk about giving a kanji a singular "meaning". Take this set of words:
着る{きる}、着物{きもの}、着く{つく}、着ける{つける}、接着{せっちゃく}、到着{とうちゃく}、着衣{ちゃくい}
In some cases 着 has a meaning "to wear". Actually, whenever it uses the kun-reading き・ぎ it has this meaning. In some cases it means "to arrive", as in つく, however the verb つける which also uses the reading つ can also mean to put on clothing, as in 身{み}に着{つ}ける. In the case of the words where it is read ちゃく it could take any of several meanings, including "wear" or "arrive".
I would rephrase: with a few exceptions, the reading of a kanji in a particular word is not linked to the meaning of the same word.
(2) A reading is a way of speaking the kanji out loud.
Fair enough.
(3) Kun readings are always used when the kanji appears alone, except apparently with numerals, and with those kanji which lack Kun readings.
(4) On readings are almost always used when the kanji appears as part of a compound word.
(3) is not true; it's a guideline, not a hard rule. Kun readings are almost always used when there is okurigana (e.g. for verbs excluding suru verbs and for i-adjectives), but there are words read with on-readings which are standalone kanji (e.g. 本{ほん}). If you add an "almost" there, as with (4), it would be right.
(5) In the vast majority of cases, there is only one correct reading for a kanji in a given context. That is to say, most compound words are spoken aloud in only one way, as are most solitary characters.
In the comments dainichiさん and chocolateさん gave some info about when this is not true, but generally, yes. A word has a reading. A 'solitary character' is just a word written with a single kanji, so it has a reading.
(6) If a kanji has multiple On readings, then there is no way to know which On reading to use for it in a compound word except to simply learn by trial and error as other people correct you, or to look up every new word you find to see how it is pronounced. Unless there are furigana.
Not just if a kanji has multiple on-readings. You can guess at a compound reading using the on-readings and be correct a fairly high percentage of the time. However, even if both kanji only have one on-reading, the compound could use kun-readings or have a special reading, and there may be 連濁{れんだく} (see here ) which isn't 100% predictable either.
(7) There are a series of different On readings called go-on, too-on, and so on (you know what I'm talking about better than I do), and many Kanji have a different reading for each one, and all are used in different contexts, and a learner must memorize all of them for most Kanji.
On-readings can be categorised by what time period they were imported from China. I've never paid much attention to which reading was which type as I don't think it makes any difference in the modern language.
I do not think that a learner should memorise every reading for every kanji, no more than someone learning English should learn all the possible ways ~ough can be pronounced. For a start, as we've established, there isn't a rule about which reading to use where, so just knowing a list of readings isn't very useful. Also, some readings might not be as common as others. It is better to learn a few common words - then you will also learn the common readings and exceptions.
Taking an example of a simple, common kanji: 白
Probably if you looked this up you would see しろ・しら as the kun-readings and はく、びゃく and the on-readings. However, しら and びゃく are not that common, and for a beginner knowing しろ as in 白{しろ}い and はく as in 告白{こくはく} is plenty; you can worry about the other readings when you start learning words that use them.
