Just to back up what you've already been told, mostly.
1) Is there any way for this agency to get round this legally and ethically?
You can, apparently, get a VAT number more or less specifically for this situation. No personal experience. I'm VAT registered in the "conventional" way, as it were.
2) As a UK firm, how burdensome is being VAT registered?
Depends how you define burdensome Takes me about 15-30 minutes every quarter, including the European Sales List. But my software spews out the figures for me, all I have to do is check them & type them on the hmrc website (and in truth, some s/ware will upload it for you too).
Do not overlook the ESL in your VAT admin if you have EU clients.
3) Would I have to charge VAT to all UK customers?
You now seem fully informed on this point
4) Would I have to charge VAT to all non-UK customers?
You seem reasonably clear on this too by now, but I'll state it the way I always understood it. HMRC's intro to VAT (notice 700) states that "supplies are outside the scope of VAT if they are.... made outside the UK". So far, so good, but I am in fact in the UK - so do I make my supply here?
Notice 741b section 8.1.3 tells us that translation is a service that is not supplied where it is performed. That being so, the "place of supply" is "where the customer belongs", so if the customer is outside the UK, notice 700 applies and translation is outside the scope of VAT. QED.
5) Are there any other benefits? I suppose I can claim VAT back on corporate purchases,
Yup. To me, a major benefit, apart from getting a few quid back over the course of the year, is avoiding the issue you started the thread with. Although there are other ways of achieving this, it seems.
Are there any major disadvantages?
I would say not. But then I do a great deal of my own number crunching. Just as well, as my accountant has proved to be somewhat less than efficient. I had to tell THEM about the Employment Allowance and the impact on the optimum director's salary. So I'm now running the payroll side myself and am gradually looking to take over the whole shooting match (filing and whatnot). I digress. Point is I'm very happy playing with figures anyway, so it's no issue to me, and if you are similar, then you might find the same.
If you let an accountant do it all for you I would assume they will charge more.
[Edited at 2015-07-01 12:04 GMT]