Yes. A limited company can pay into a director's pension and deduct it as a business expense. No National Insurance is due on it, nothing is taxed on you, and the limit on your own earnings does not apply to it.
A dividend is paid out of profit that has already borne corporation tax, and is then taxed again in your hands. A pension contribution is neither.
No. A pension contribution is an expense, so it can create or increase a loss. Whether that is sensible is a separate question for your accountant.
Carry forward needs you to have been a member of a registered pension scheme in each earlier year you want to use. It does not require earnings in those years, and it does not require the company to have existed.
Not for pension purposes beyond the allowances. For corporation tax, where a contribution exceeds 110% of the previous period's by £500,000 or more, the relief is spread across future periods.
Your own contributions would be capped near that figure. The company's are not, which is why the route matters for directors who pay themselves mostly in dividends.
The figure depends on your allowance, the taper and what you have carried forward. Settle it well before your year end.
A first conversation of about twenty minutes, at no cost to you. Tell us what is on your mind: a pension you have lost track of, a fixed rate ending, a will you keep meaning to write.