a) Times New Roman is very rare: you're either confusing the font with a different one, or you were looking at themes whose fonts aren't installed in your computer (that's also rare, because most themes use common fonts). See list here:
b) Apparently WP has changed something, so what I was thinking of is also possible in the theme you're using. Paste this code in a text widget:
<div align="center" style="width:100%;top:0;left:0;position:absolute;z-index:11;">
<a style="border:none;" href="mailto:EMAIL ADDRESS HERE">
This code will create an invisible linking rectangle in front of your header e-mail address.
c) If you decide to change theme, this sort of code (with different numbers) will work well in the first two groups of themes listed here:
Combine it with the list in the following post, to exclude themes with no header image:
In the same list you'll notice that many themes allow you to hide the regular blog title and tagline. So if I were you I would switch to one of those themes, create a new header image (with the logo, the tagline and all the rest), and take advantage of the hide-text option to avoid the redundancy you've got now.