Alright, I'll go for the rest.
In the first place, I have to point out that what you asked is far more complicated probably than you thought. There are many types of compound verbs in Japanese. Some compound verbs just connect the two original meanings of each verb (e.g 走り去る, 焼き切る), and some have little to do with the original meanings and can be considered as a new verb (e.g. 打ち解ける, 引き立つ). Also, there are ones that only V1, which is called 前項動詞 "first compound verb" in Japanese grammar, gives the main meaning, and V2 (後項動詞 "second compound verb") only adds grammatical information such like aspect, direction of the movement or emphasis. In these cases, V2 can be considered a kind of auxiliary verb, and whether the compound verb is intransitive(Vi) or transitive(Vt) is decided by V1, as same as verbs attached to the -te form such like いる/ある/おく, いく/くる or やる/もらう/くれる. (Incidentally, these verbs are called 補助動詞, and the most common English translation of this is "auxiliary verb").
e.g.
始める (Vt, "the action starts")
降る Vi --> 降り始める Vi
書く Vt --> 書き始める Vt
過ぎる (Vi, "to do too much")
動く Vi --> 動き過ぎる Vi
動かす Vt --> 動かし過ぎる Vt
However, there are also verbs that can be considered as an auxiliary verb from the perspective of function, but also rules the intransitivity/transitivity of the compound verb.
e.g.
かかる (Vi, "direction of action towards~")
飛ぶ Vi --> 飛びかかる Vi
殴る Vt --> 殴りかかる Vi
And there also exists another pattern that V2 differs depending on V1.
e.g.
合う / 合わせる (Vi / Vt, adding a nuance of mutual)
混ざる Vi --> 混ざり合う Vi
混ぜる Vt --> 混ぜ合わせる Vt
(Note that 混ぜ合う (Vt) also exists. The meanings are "Two things mix", "The agent mixes two things" and "Two agents do the action 'mix' together", respectively.)
Furthermore, there are ones that V2 is the main and V1 works like a prefix (e.g. 差す(Vt); 差し迫る (Vi), 差し出す(Vt)), or Vi and Vt are the same form (e.g. 引く(Vt) 上げる(Vt); 敵軍が国境から引き上げた (Vi), 政府は軍を紛争地帯から引き上げた (Vt)).
Now you can see that exceptions exist even for 1) and 2), right?