If you have the Genki books, you can find this construction on Genki I, Chapter 11, Grammar Point 1 (with 〜たい). The translation of 〜たがている is more like "it seems X wants to Y" because the 〜たい construction is only for personal "wants", not someone else's. As my teacher explained it, because it had that "speaking for someone else" intent, it would be used in situations like a parent speaking for a child or where the person is deducing the intent of someone else's actions.

When you A したがる, you are showing out that you want to do A. したい（adjective conjugation) →　した・がる 犬がご飯を食べたがっている。 When you B がる, you are showing out that you are in the state of B. ネイマールが痛がっている。 子どもがおもちゃを欲しがっている。 The essence is "showing out". 子どもがおもちゃを欲しがっている。 You can recognize their desire for toys because they are showing it out by their expression or behavior, which you can see with your eyes. You cannot recognize it when they keep their desire in mind and don't show it out. NG: 私は水を飲みたがっている。 Basically you cannot use たがる to describe yourself because you cannot see yourself. OK: 私はさびしい。 NG: 私はさびしがる。 However, the following expressions are possible, OK: 鏡の中の私は、もっと、やせたがっているように見えた。 because you can see yourself in the mirror. OK: 私は彼に、いやがって見せた。 because he can see you. OK: 私は、私の体が水を飲みたがっているのがわかる。 because you can see your body, or you regard your body as an independent object in this sentence. >1) What is the rule for this conjugation so that I can apply it to other verbs? Before がる, there has to be placed an adjective conjugation, (not a verb). (し）たい　→　したがる ほしい　→　ほしがる いたい　→　いたがる うれしい　→　うれしがる いやだ　→　いやがる かわいい　→　かわいがる