In modern standard Japanese, はなし as a noun is always written as 話, without し. This rule is taught at school today and is fairly strict, although you may see exceptions in old documents. As a verb, はなす is written with okurigana.
話 is not a suru-verb. You usually need a direct object marker を when it's used with する.
シンデレラの話【はなし】をする。 (話 is a noun)
はなしして and はなしした can appear in two situations:
When はなし is a noun and the following を is omitted because it's a casual sentence.
ママ、シンデレラの話【はなし】して! (話 is a noun)
As part of a humble expression お + masu-stem + する
.
本日は、世界の童話についてお話【はな】ししていきます。
To distinguish, first note that omission of を after 話 can appear only in fairly casual conversations. And looking at the modifiers will usually tell whether it's used as a noun or a verb. Unless 話 is clearly modified by an adjectival expression in a casual conversational sentence, you can assume 話して/話した is a verb. In your examples, 「話しながら」「何人に話した」「彼は大声で話した」 are theoretically ambiguous, but you can usually assume these はなし are verbs.
- 先生が楽しい話【はなし】してるよ! (a noun follows after 楽しい)
- 先生は楽しく話【はな】している。 (a verb follows after 楽しく)
Humble おはなしします should be written as お話しします, because 話す is a verb in this construction (cf. お預かりします, お持ちしましょう). That means your third expression is grammatically wrong, although this mistake is found even among native speakers.