What a timely question.
Does DNA contain information beyond protein synthesis?
Yes. In fact, protein-coding genes only constitute a tiny part – less than 2% – of the whole DNA. There are of course many other genes which aren’t protein coding: there are genes for ribosomal RNA and we find more and more genes which code for small RNAs, such as tRNA. But even if we count all those genes we won’t come above maybe 10% of the total DNA.
Most of the DNA is instead devoted to the regulation of gene expression, most importantly via the binding of transcription factors (but the picture is much more complex than that). With the conclusion of the ENCODE project, a whole slew of papers were published which show that in fact most of the DNA is actively implicated in the binding of various factors (although it’s not known how much of that actually contributed to the cell’s fitness).
But I’ve hijacked your question a bit here. So let’s come back to what you’re actually interested in:
What about higher levels, such as organelles, cells, tissue, organs?
There is no known mechanism (beyond the already mentioned regulation) which would encode such information in the DNA. Excluding its existence categorically may be hard but given that we haven’t found any machinery which would be necessary to read such information, we can be pretty confident that it doesn’t exist.
If not, what guides those levels of structure?
The higher levels of organisation are to all appearances emergent. That is, they are a consequence of the lower level organisation. For instance, take the cytoskeleton which carries much of the cell’s physical structure. It is composed of different protein complexes which form spontaneously through assembly of globular proteins (such as actin). The are several ways in which the process can be guided but strikingly it’s largely stochastic – i.e. mostly unguided, and it still succeeds in building a stable skeleton, simply by virtue of molecular properties encoded in the proteins.
I think this is a common theme of cell organisation: the elementary building blocks are encoded by the DNA, and their abundance is tightly regulated. Everything else, i.e. higher-level organisation – follows from that: abundance and localisation of the right proteins.
As for differentation into different types of cells, and organization into tissue and organs: the information and control systems for that are currently unknown.
In fact, much is known here, and it goes back to regulation on the level of DNA: we know that genes are differentially expressed depending on the cell type and stage of development (and the stage in the cell cycle). This regulation is highly complex and decoding it is a slow process. Nevertheless, the factors involved here are decoded one by one. This is the domain of developmental biology.