First I want you to know you are asking about what people would call the difference between on-reading and kun-reading (音読み versus 訓読み), which are ancient chinese (or sino-japanese, as my professor refers it) reading and (traditional) japanese reading respectively. Simply speaking, chinese originated words (or parts) are pronounced in sino-Japanese ways. japanese originated words are pronounced in traditional Japanese ways. and that was supposed to be the grand, master, iron rule of pronouncing things in Japanese
Second is, the state of Japanese is defined on a "sort of as-is" basis, that is to say, while there are some unclear rules, whatever people are used to reading in practice, it may become regulation. This is why you see these different readings. Rendaku(連濁, or sequential voicing, another thing that might change the pronunciation of a word) is a wonderful example of a twisted (metaphorically) morphology rule in Japanese. Japanese people sometimes deliberately play with the pronunciation of a kanji to express even more meanings than it already have.
So what I would recommend is, do it the hard way, learn a lot of words, before you start guessing.(as opposed to trying to find a general rule)
Back to your answer. I think you sort of got it in your two rules, they are two special cases of two meta-rules.
1, prefix can be associated to an on-reading because words that are pronounced in sino-japanese are either of chinese origin, or japanese-made-chinese word.
This is why you feel they are always prefix of something: most chinese words have two chinese charecters.
2, On the opposite, when a kanji stands alone as a word or with a kana on its side, it is more likely a kun-reading word. "hokano hito" is a good example because the kanji 他 is alone. But relating it with the use of "no" might not be such a good idea. because no connects nouns, and nouns are.... nasty.
Again, there are many exceptions to my rule 1 and 2, I know you guys are gonna list it any way, so i might as well provide some:
To your second question: not all kanji can distinguish meaning with its pronunciation alone in a word. and unfortunately i find it hard to do so with 他.
You can often tell the origin and sound changes happened along the history though, which nobody cares unless you are linguist or really, really love the language.
btw there are more readings of this kanji that are not used, thought you might be interested. from http://kotobank.jp/word/%E4%BB%96
［音］タ（呉）（漢） ［訓］ほか あだし
［名のり］(<-that's for people's names)おさ・ひと
I would definitely read 他の物 "hokano mono" because its less confusing. ta+no become the head/stem of words such as "tano-mu" "tano-mashii", which both have a tail started with m.