I only just caught up with this thread, so I apologise if I am repeating anything, but I can speak from (UK resident) experience.
You do not need to be a limited company to register for VAT. Being a sole trader is fine and has advantages in that tax returns are easy and can be done by yourself, while those for Limited Companies are more complex and really do need a professional advisor to ensure they are correct. £1000 for the first year is a bit steep in the current market, but not by too much.
The difficulty is persuading HMRC to allow you to register for VAT. We needed a letter from the flying school saying that he was completing a Commercial course, and that there was good employment opportunities with this qualification (it was a few years ago!). We could then claim back the VAT paid to the training school every three months. We also built up a significant trading loss (obviously) in that period which we entered as part of the annual the self assessment on-line tax return. The rules are that you can carry forward losses from previous years to offset profit in subsequent years so, for the first year or so when there were significant earnings, this reduced the tax liability to zero. However, there is a time limit on how long you can do this for.
Being VAT registered, you obviously need to charge VAT on your invoices (assuming you have stayed self employed) which you then offset against any VAT you have been charged on your expenses (OPC etc), and the rest you pay back to HMRC every three months. I recommend the Fixed Rate VAT scheme which saves bureaucracy and is normally financially beneficial also.
Finally keep very good records including every item of spend and income (spreadsheets are fine) and then, if you are unlucky enough to get a visit from HMRC, you should have no problem. The HMRC web site has loads of advice and guidance on all of these issues, so, like always, do your homework first.
Unfortunately, I cannot speak for overseas students.