What do you mean by exist ? Do real numbers exist ?

With sufficient technology could we take advantage of higher dimensions for travel or other purposes, for instance.

Oh you are mixing Physics and Maths. You are asking whether the space we live in has many dimensions, this article about is about the mathematics of spaces that does have many dimensions. One can use the mathematics of such spaces to, for example, find stored images nearest to a query image, nearest document to a query document and so on.

Oh yes, they do :)

Don't know why I got downvoted, it was a serious question.

Can you give an example of how we've made use of the 4th, 5th, 6th dimension, etc.?

Higher dimensional geometry can show up in lower dimensional problems. This numberphile video (https://www.youtube.com/watch?v=6_yU9eJ0NxA) involves a puzzle about throwing darts at a dart board which is solved by using the volumes of 4+ dimensional spheres.

doesn't 4th just include time with a 3d object. Would things like waves/pulses fall under that? 5th seems to be used in the Kaluza-Klein theory[0] for gravitation and electromagnetism.

[0] https://en.m.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory

edit: I agree, I don't know why you got down voted. It was a thought provoking question.