Op, my DH runs a limited company, with his accountant doing most of financial work.
I dont generally have or need access to his business accounts but I do know who the accountant is, and who he banks with, and he would show me bank statements if I asked. Its called being sensible, as you say, if anything happens to him, you need to at least be able to contact the accountant to wind the business up.
As for tax form, well my DH generally does his own, but is aware of what extra tax he may need to pay - as a pp said, accountants are on top of this, so he is usually told the exact figures to put on his tax form by his accountant. In fact he is often contacted earlier in the year, if near a change in tax threshold, and the accountant suggests more goes into pension, or less dividends etc.
This year the accountant did tax forms for both of us as we had some CGT to pay. I've never done one before as I am employed and on PAYE. He told both of us ahead of time how much tax we would owe based on our earnings and the CGT calcs.
So, either your DH has a crap accountant - or none at all. I do wonder why he wont even tell you their name? What is he hiding there?