Crossing situation in the TSS. Which vessel is give way and by which rule and actions for both the vessels?

When dealing with TSS situations, always keep this in mind.

A vessel following a Traffic separation scheme does not get any special rights over any other vessel.

So for all the TSS situations consider that if there is a risk of collision, the rules for that situation would apply.

In the case of crossing situation in TSS, the rule 15 of the COLREGS would apply.