Is it incorrect to use 儿化 (érhuà) inside a word, e.g., 门儿口, and 小孩儿子 and 玩儿具?
"门儿口", "小孩儿子" and "玩儿具" are all incorrect, we don't say that. In the final analysis this is an idiomatic usage issue.

For example we say 东便门儿, 西便门儿; but we don't say "天安门儿". (There's a historical reason behind.)

Particularly, 小孩儿 and 小孩子 have similar meaning and usage, "小孩儿子" sounds repetitive and redundant. 玩儿 is usually used as a verb, you can say 玩儿游戏, 玩儿手机, 玩儿玩具; but not 玩儿具, 古玩儿, 玩儿笑.


  Is it incorrect to use 儿化 inside a word?


Yes you can, but only in the specified cases; i.e. in which conforming to the idiomatic usages. e.g. 锅儿挑, 巴儿狗.

It is known that it's hard to summarize logical rules for 儿化音. This 北京话儿化词典.pdf might help you.