Your problem might have been fixed by changing divs to spans. But what you describe shouldn't really happen because of it.
I would guess (since you really do not want us to provide thoughtful suggestions but wild guesses) that the problem is caused by the table.
Tables are rendered different across the browsers. But the main character of a table is that it expands to fit the data inside. So browsers will load a table and make sure that everything fits into it. However, with large tables that have a lot of data, some browsers start rendering the table before they receive all the data within it. If you have a really long value at the very end of the table (one that will extend a certain column), the whole table will need to change according to that column change.
Here's the issue. If the browser waits until all the data is loaded before it outputs the table, your error shouldn't happen. Apparently this is how most browsers work. However, FF wants to render the table while it is still being loaded. If half way through FF felt the table was 80% wide, it will render it at that. If at the end you will have a really long value, this will get corrected. However, if you will have floated elements, it will not, because floated elements do not want to extend the parent, but simply stack below. Clearly FF is quick enough on the local computer to display the whole table the way it should. On the remote server, some elements take longer to load, so FF outputs interim table, which is not of the same proportions as you want it, and when you add floated elements, they don't push it to the expected proportions anymore and the page fails.
This is not a bug, it is more the way the tables behave. Given that most of the time tables contain nothing but simple tabular data in the form of simple text and numbers, the issue is not really apparent -- the data loads fast enough and values are either decided to break if there's not enough space, or change the proportions of the whole table.
If you use tables for layout, these errors can occur. I have seen them occur in most browsers, don't think it's limited just to FF. There are only two ways of avoiding it:
1. Don't use tables for layout.
2. Strictly define the table width (and the width of all the columns) at the beginning of the tables (col tags for example) and use fixed table layout to force proportions to be the way you want them to be.
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