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Actually, the original root was tRR (the dental t with a long sonorant r). It is a very old Indoeuropean root ultimately cognate with words like Latin trans, and meant to pass or live through or over till the end. An important metaphor for this world as an ocean in which one can swim or sink shaped many of the semantic shifts of this world: so the swimming it refers to is distinctly shaped in many senses with a sense of being saved (thus the poster's `also possibly emerging' above). Thus taran (with a schwa) means swimming, whereas tAran (with a low back vowel; technically the grade of the vowel is changed due to a causative) means saving.

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The original meaning of passing through or across (the primary meaning in the Rigveda) is lost in later usage where it also became the ordinary word for swimming and floating, as opposed to sinking and dying.

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tAl is not Sanskrit, but taD.Aga (the D. is a rhotacized retroflex voiced unaspirated sound: in the intervocalic position in many dialects a retroflex l has long been an allophone) and related words (a for A, k for g) are, and they mean a pool (and the ultimate origin of the English tank). The etymology is uncertain. I am not sure whether Hindi tAl originates here, or whether there is a Persian or some such origin.