The "s you see in the HTML attribute are character references for double quotes. They are represented in a CSS attribute selector as literal double quotes, not the HTML entities, since HTML entities pertain to HTML only. Since the value contains double quotes, use single quotes to delimit the value in your attribute selector so you don't have to escape the double quotes in the value.
The backslashes in the attribute value need to be escaped, so \ becomes \\.
There is also a wayward leading space in your attribute selector that should be removed.
Hence:
[data-store='{"dialogURI":"\\/messages\\/compose\\/dialog\\/"}'] {
position:fixed!important;
bottom:0px!important;
}
As mentioned by others, if the element has a class and you can identify that element using that class and some context, it's a better idea to do that. However, the element in question may not always appear in the same context or may not be the only one with the class, and you don't always have control over the markup — so there is nothing wrong with using an attribute selector if you don't have any other options.