As @snailboat and @Chocolate have noted, I think you are actually thinking about something called 複合動詞{ふくごうどうし} (or compound verbs.)
It is interesting to consider combining more than two verbs when creating 複合動詞{ふくごうどうし}... and it certainly is possible, according to the example word lists given in this paper by 林 翠芳 (LIN Cuifang). LIN gives examples of times where 複合動詞{ふくごうどうし} can involve the process of combining three separate verbs into one compound verb (三次結合複合動詞{さんじけつごうふくごうどうし} vs 二次結合複合動詞{にじけつごうふくごうどうし}).
For example:
書{か}く+立{た}てる+すぎる = 書{か}き+立{た}て+すぎる = 書{か}き立{た}てすぎる
立{た}つ+止{と}まる+かける = 立{た}ち+止{ど}まり+かける = 立{た}ち止{ど}まりかける
So in answer to the first question, while there are times where three-verb-combined verbs exist, it's possible that the reason two-verb-combined verbs are more commonly seen is because the words have a simpler meaning or definition. In other words, the more convoluted a word becomes, the more complex its meaning.
In answer to the second question, there has been at least one similar question asked in regard to the meaning of ー込{こ}む when used at the end of a verb. Though the meaning may at times go beyond what is listed in the current list of answers for that other question, I would still recommend reading that other question in order to get a better idea of any additional subtleties related to ー込{こ}む.
Finally, verbs ending with ー回{まわ}る are fairly straightforward (typically being related to something "around" or "about", as you noted.)
As far as 複合動詞{ふくごうどうし} go in general, I would recommend taking a look at this paper when time permits. The paper better explains different functions, meanings, and aspects of 複合動詞{ふくごうどうし}.