Re Q2,

MR0374131 (51 #10331) Reviewed <br>
Hironaka, Heisuke <br>
Triangulations of algebraic sets. Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, <br> Humboldt State Univ., Arcata, Calif., 1974), pp. 165–185. Amer. Math. Soc., Providence, R.I., 1975. <br>
14B99 (32B20 57C15) <br>

In case you are wondering: no, this does not use resolution of singularities.  Hironaka's point in writing the article was to demonstrate that resolution is unnecessary for triangulation.