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I believe the answer to your last two questions is yes, and it follows from Litherland's (1979) computation of the Tristram-Levigne Tristram-Levine invariants of torus knots. See Kearton's survey here:

http://www.maths.ed.ac.uk/~aar/slides/durham.pdf

i.e. a connect sum of torus knots is slice if and only if they the prime summands appear in balancing mirror-reflected pairs.

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I believe the answer to your last two questions is yes, and it follows from Litherland's (1979) computation of the Tristram-Levigne invariants of torus knots. See Kearton's survey here:

http://www.maths.ed.ac.uk/~aar/slides/durham.pdf

i.e. a connect sum of torus knots is slice if and only if they appear in balancing mirror-reflected pairs.