How do I have the the name and email fields to be next to one another?
You could consider using Contact Form 7 Skins to style your Contact Form 7 Forms. You can now use the add-on CF7 Skins Ready to put name and email fields next to each other
See also the following CF7 Skins Documentation which might help you:
Contact Form 7 fields side by side – shows how to place form fields side by side on a single line using the CF7 Skins Ready Add-on
I have attempted to adjust the height of the name and email field but nothing is happening. Do I use px or %?
See Styling Contact Form 7 Forms for a general explanation of styling CF7 forms using CSS.
If you find you need more detailed advice, after reading and working through the above link, please include a link to your Contact Form 7 form, so others here can examine your form in detail using Firebug or Chrome Dev Tools to understand the CSS used for your CF7 form elements and provide you with a possible solution.