CSS and PHP in Stylesheet and Header.php File (strange problem)

First of all, here is the header code I’m working with:

[Huge chunk of code and CSS moderated as per the Forum Rules. Please use the pastebin]

Here is my website: https://www.velociteen.com

The problem: I need to figure out how to effectively replace that live chat button into my template so it looks nicer (maybe move it over to the right of the Header image or something) and I need to move my content up higher towards the navigation bar so there is only a few pixels worth of problem; have been trying to solve this problem for the past few hours and I can’t come up with any solutions.

Any ideas?

Thanks!