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1. I think you have two options here. Firstly, on a quick look, I have seen Eon's major operations are in the energy infrastructure and distribution space. You could thus research what types of disputes work this normally entails, as this is the area the company's litigation team will likely spend most time in. Secondly, the company should be aware of the fact that information about their legal work is not easily available online, and therefore will likely not expect you to know much. If you want to further hedge against that risk, you could ask a question phrased something like 'I have a broad interest in commercial litigation, but could you please let me know what types of matters the team is working on most at the moment, so that I can tell you what I am most interested in regards to that?' If phrased this way, the question no longer makes it seem like you do not know what legal work they do in general, but just that you do not know what is keeping them busy at this particular moment, which I cannot imagine they will have expected you to know.
2. Unfortunately, it is quite common for firms to take a couple of days or even weeks after their stated deadline before coming back to applicants. If that is the case, the reason is that they are quite busy, and therefore, they may not even have time to respond to your email. That said, there's no real downside to doing it if you word it in a polite manner (although I would perhaps wait a few more days). I would go for something like:
Hi/Dear person's name/Graduate Recruitment Team,
My name is ... and I am writing to ask about my application to the firm's summer vacation scheme (include application reference number if you have one). I appreciate that you will be very busy at the moment, but in light of other professional opportunities and to avoid potential scheduling conflicts, I wanted to ask around what time I could expect to hear back with an update on my application.
Many thanks,
....
