I have a semi-related question(s): We have shown that if b^{2} - 4c < 0,then x^{2} + bx + c > 0 for all x.

Now, if I turn around and say the "reverse": If x^{2} + bx + c > 0 for all x, then b^{2} - 4c < 0.

Question 1: Is what I have just written called the "converse" of the original statement? And I do not think that in general the converse follows .... I would have to prove it. Correct?

Yes, that's the converse of the original statement, and it is true that the converse does not necessarily have to be true when the original statement is true.

Here's a simple example:

If x = -2, then x^{2} = 4

The converse is: If x^{2} = 4, then x = -2, which is not true.