Would using LaTeX instead of LibreOffice benefit me for writing my dissertation?

I really hope you do find the time to learn LaTeX. You'll never look at documents the same way again. It can really turn a person into a bit of a document snob :)

LaTeX is not a word processor. You cannot 'export' to Word. LaTex is for professional quality document processing. MS Word is for cramming some text together as quickly and conveniently as possible for the writer. LaTeX is for deliberate, scientific, artistic and careful design for optimal readability. I'm not saying that word processors are bad! They are two different tools for two different jobs. Check out The Beauty of LaTeX, to see illustrations some differences.

LaTeX will give you a pdf, postscript or dvi file, which are not editable by MS Word. Nor would you want to do so, even if it were possible.

LaTeX has no problem with images, as long as you are using a modern driver as in PDFLaTeX. For some idea regarding what LaTeX can do, check out the TeX showcase

LaTeX (or rather BiBTeX) will generate your bibliography based on a database of information. I recommend you check out Mendeley for managing your references. It works brilliantly with LaTeX and LibreOffice.

LaTeX will also handle your table of contents, list of figures, list of tables, list of acronyms and index.

While I do recommend you learn LaTeX, I don't recommend that you write your dissertation in LaTeX with the intent to submit as a MS Word file.

The notion of an academic institution demanding a dissertation in MS Word format is unethical---even scandalous. If my University had any such requirement for a non-open standard, I would raise a hell the likes of which they had never seen. Needless to say, I would refuse to submit in a non-open standard format.

As others, I also believe that have no sense for a novice to learn LaTeX if the final target is Word. However, if you are an experienced LaTeX user, probably you will not desire to work too much with bloat-ware word processors and probably you have already something made in LaTeX. Only then exporting could be a good idea.

Is it easy to export to an MS Word format?

Yes. There are tools like latex2rtf, mk4ht or elyxer to convert to ODT, RTF and HTML formats that can be read by LibreOffice and saved in Word format.

For a novice a easy option could be write the documents in LyX or import LaTeX files to LyX because you can export to the above formats trough a simple menu, that internally call to the appropriate exporting tools. (Moreover, it a good option for a novice because you can write basic LaTeX documents without previous knowledge of LaTeX code).

However, the conversion is far from perfection. This could be painless for large but simple texts as hopefully is maintained the basic style typography (italics, etc.) and document structure (table of contents, sections, cite references, etc.) and lost format (margins and headers in HTML export, for example) are easy to fix later in Libreoffice.

For more complex documents (for example with a lot of equations, tables or figures) this final adjust could be more laborious. If it is also a very customized document (large preamble with uncommon packages and custom macros, or a lot of formatting code within the body text) then you can expect more serious problem (special formats are lost, or exported as garbage code, or you can export anything because the exporting tools do not know how manage the complex document). You mileage may vary, but the more complex the LaTeX source, the less chances of success.

Apparently support for images is "flaky", can anyone quantify this, or has it improved?

In what sense flaky?

First, you must understand that LaTeX is not kind of word processor with menu to insert images that can be placed or scaled with the mouse. LaTeX is simply you writing plain text with any text editor (even the simple Notepad from Windows is enough) that later can be converted into a formatted PDF (or DVI) with a compiler as pdflatex (there are more options). Lyx and many text editors can help writing the most common LaTeX codes for you, including the basic code to include an image, but these programs cannot cover all the possibilities of LaTeX with graphics. Understood this, what I can do with images writing only plain text?

• Include external images in differents formats? Yes, you can include the common PDF, PNG, JPEG and GIF images. With LyX even SVG (that are converted to PNG). Need more? There are excellent tools to bulk convert other format images.
• Rotate or rotating the images? Yes, you can.
• Scaling the images? Yes, you can.
• Find a good/elegant position for images? Yes. In fact, the float figures in LaTeX was for me one main reason to left Word/OpenOffice, where positioning of images is rather time-consuming and frustrating even when you are well experienced about arrangement, anchoring and text wrapping. But moreover, you can do much more that using standard floats (read about the Tufte format, for example).
• Draw directly vectorial images, diagrams? Yes, you can. Even an animated chicken hatched from an egg. See How can I draw an egg using TikZ?
• Draw plots from data? Yes you can. Directly with pgfplots package or indirectly, including chunks of R code (statistics language) or LaTeX code generated with Gnuplot, for example.
• Write text or paint marks over the images? Yes, you can.
• Round borders? Yes, you can. See LaTeX Photo With Rounded Corners
• Any other thing that I can imagine about images? Probably, although surely you can do better or faster with more appropriate tools than an high-quality typesetting system. After all, LaTeX was not designed to compete with Inkscape or The Gimp.

Can anyone think of anything I have potentially overlooked for writing such a large document?

The possibilities of format that offers a markup language like LaTeX are incredible but a elegant default format for simple text is obtained simply with the first line of code defining the document class:

   \documentclass{article}

This basic format can be extended or modified in the preamble, (from this line until \begin{document}) to include images for example, but then you can mostly focus on the content. Although the content is not just clean text, since it must usually have many commands about the text structure, as \section{Some text} (instruct to the compiler that "Some Text" must be formatted as the title of a numbered section). This is hard to manage at first, but when you learn the most common commands, the hard thing is to leave your favourite lightweight text editor to return to LibreOffice or Word.

I learned the LaTeX basics this year and liked the output it produces. However, there are two main reasons that threw me back to MS Word:

1. if you work in a team and your collaborators don't use LaTeX and don't like PDF, you're buggered. Especially if you send out drafts of chapters of your work, you won't get the kind of feedback you hope for if your commentators can't use their word processor of choice, which in my field unfortunately always is Word.
2. I found LaTeX to be less supportive of the writing process than I hoped for, which is mainly due to the lack of possibilities to annotate your own text with highlights and comments. Any other word processor comes in more handy in this respect, although I don't want to rule out that that's just due to my insufficient understanding of LaTeX. However, if you want to revise your documents regularly (which you should), you'll find a Word-like processor more efficient, probably.