I know the translation of the sentence, but I don't understand some structures. Is もし from a verb? And what does いくら mean here?
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You are parsing it wrong.
車を運転する時は means "when you drive a car" and explains in what circumstances the latter part of the sentence applies.
いくら = how much to / to what degree
注意 has two meanings. Either to warn someone or to be mindful of something.
して = て-form of する
も - this makes it a pattern いくらVても - "no matter how much". In this case, roughly "how careful you are"
しすぎる = し is the stem of する and すぎる is to do something too much. ~ことはない = there is no such thing as ~.
So for the whole:
When driving a car, there is no such thing as being too careful.
When you're driving a car, you can never be too careful.