They're sometimes interchangeable but not always. Here's a good explanation from Merriam-Webster:
'Since' vs. 'As' vs. 'Because'
"Because" is the most common and most generally usable. "As" sounds quite formal and is uncommon in casual conversation.
As the linked article explained, "since" has a weaker sense of causality than "because." For that reason,
Yesterday I was late for work since my car broke down.
sounds a little odd to me.
In the third sentence, "as" is not usable because the sentence would be understood to mean, "The children stopped playing football at the time it started to rain."
"Since" and "as" are often more comfortable in a reverse construction:
As it was raining hard, the children were unable to play football.
Note the formal register, which goes well with "as" as a conjunction.
When in doubt, use "because." And don't forget about "so" or the null conjunction. Both of these sound more natural to me than any of the three conjunctions you mentioned:
Tom couldn't sleep at all, so he got up in the middle of the night.
Tom got up in the middle of the night. He couldn't sleep at all.