Some questions which will help people answer you:
1. What's your broad field/subject area?
2. Which country/countries are you applying in? Are you a home or international student?

Ahhh. :)

Well clearly the first sounds more correct if read aloud (or in one's head). And is the one I'd be inclined to go for. I'm pretty sure that some people write things like 'An historic event' and get away with it, even though that only makes sense if you drop the 'h' entirely.

I'll be honest, I'm not entirely sure if there's a rule exception for this one. I'd suggest finding the most pedantic reviewer you know and asking them. They may be wrong, but ultimately they're the ones you would have to get it past!

Congratulations!

Well I'd strongly recommend googling 'critical appraisal of scientific literature' to get a more in depth perspective, but in brief, there are lots of ways a paper/study can be bad e.g.
- if it has a poorly defined question
- if it uses an inappropriate method to test the hypothesis
- if the method has been incorrectly applied
- if the interpretation wildly overstates the importance of the findings
- if the results are misinterpreted
- if the stats are manipulated to try and salvage something from the wreckage of a poor study or one which would be more honestly reported as negative.
- if it ignores the current understanding of the topic

Some of those occur more rarely than others. I'd say over/mis-interpretation of the results might be the most common.

I'd get in touch. A lot of senior staff can take a long time to reply to emails, especially if from new sources. I think the wording you suggest is probably fine. Maybe take the opportunity to add some extra detail about yourself, or emphasise how keen you are.

Also, if the Prof has a PA/secretary, consider getting in touch with them first, or copying them in. They often manage the email account.

I've never used either of those tools but... are you sure you need to look at them all? Is it worth developing a more refined search to reduce the number of documents?

Those are two rather different areas! It depends on your long term goals. Where do you see yourself in 5 years time? 10 years? Carrying on the legal field might lead to better job prospects, but it depends on what you want out of life.

A couple of Thesis Whisperer posts which might be relevant

I'm sure there will be training at your uni, or maybe via research councils or Vitae which can guide you on this. However, if a job description asks for experience of making grant applications, they're probably seeking people who have successfully secured funding, rather than been on a course. So if the opportunity arises to get some actual experience, take it.

I recently marked an essay competition with two other judges. We each had to give every essay a score out of ten according to a mark scheme, but deduct one mark if it was over the word count.That meant that if an essay was too long, the maximum it could score from the three judges was a total of 27 out of 30, no matter how brilliant it was.

The highest score I awarded was 9.5, but that person had gone over the word limit so it became 8.5. The winning entry had scored a 9 from me. And it was well below the word limit.

So, you can risk it if you like, but you might get penalised so that your submission is not accepted. Or they might just read the first 5000 words and stop there (electronic submissions may just cut it off). The electronic system might not even work if you're over the limit.

Ask someone for help with this as they may be a bit more detached about what can go.

Yes, they can reject your paper for not adhering to guidelines. Consider the reviewers' perspective.