Does Mathematica 7 support lazy evaluation?

It's late so only a short answer: investigate Hold[] and its relatives. With them you can implement lazy evaluating functions. Most intrinsic Mathematica functions are not lazy, a few are. In general, as a beginner, you should avoid modifying the behaviour of Mathematica's intrinsic functions, though it is very good fun to do so and can very easily make the entire system unusable.

You can solve this problem by building up the list of diagonal minors by yourself and then applying Det, for a matrix M:

Map[Det,Drop[Transpose[Drop[M,{#}]],{#}]& /@ Range[1,Dimensions[M][[1]]]]

This is a bit of a cludge but it is about 50 times faster than using Mathematica's built in Minors and picking off just the diagonal elements (tested on 100x100 random matrices).