WYSIWYG and TinyMCE stripping out float: left
My solution:

I downloaded a new module called WYSWIWYG Filter. On my Text Format, I turned off the Limit allowed HTML tags option and instead enabled the WYSIWYG Filter option. Then below in the WYSIWYG Filter settings, I configured the allowed HTML elements like so:

a[!href|target&lt;_blank|title],
div[align&lt;center?justify?left?right],
p[align&lt;center?justify?left?right],
br,span,em,strong,cite,code,blockquote,ul,ol,li,dl,dt,dd,
img[!src|alt|title|style|width|height|typeof],
@[style|title]


And I checked all the boxes to allow margins, padding, floating, etc.