1) How should we respond to those who would try to argue that the New Theology is a legitimate development in theological precision?
Show them an example of where the new 'theology' contradicts previous statements of the magisterium (which, by the way, includes the common teachings of the theologians).
2) What if they do not buy basic logic?
This makes it rather difficult, as one cannot usually give a crash course in logic to such people.
3) How do we argue against historicism?
Historicism is basically relativism: it was true then, but not now, etc. Ask if 2+2 is true now, and if it has always been so, then move on to an explanation of how that which is true is always true - and that time has ZERO to do with it.
4) Were any modern philosophers aware that they were formulating hypotheses diametrically opposed to the Catholic faith (as opposed to the invincible ignorance of Plato and Aristotle)?
Yes, but many of them were quite hostile to the Church already, such as Kant, Hegel, etc.
5) Why would modernist theologians just throw away St. Thomas Aquinas? What did they have against him? (I imagine it has to do with our fallen nature and wanting to be free from God and to not be held by fixed dogmatic definitions and formulas at the same time.)
They HATE his precision. He always clearly defines his terms, and the modernist hates being so bound by terminology.
6) Could the term "living Tradition" ever be redefined and made to be equivalent in meaning to "unchanging Tradition" in that we as Catholics live it throughout the ages?
Imo, it is not necessary, as what it means is that Tradition is the Living Voice of God (or rather, the Voice of the Living God) among men - He is always the same, and so is His voice. Those who hear it, listen.
The modern mind does not (and cannot) understand the definition, the converted mind does not need to have the definition changed.