UK Limited Company Info/Accounts

Hi All I have a few questions regarding limited companies; there is a lot of web sites that claim they can have ltd companies registered quickly, their services range from just ltd name registration (cheap) to complete company set up (expensive). a) does a company need to have a business bank account to start trading? b) can these web sites really register business bank accounts (if so, how? Surely banks need consent for credit checks and proofs of id/address to open an account??) c) if an individual registers a domain name and he or she is a company director, can ltd company trade on that website, or would the domain ownership need to be transferred to the company? d)) once the company name is registered, does the company need to be vat registered to start trading? e) if an already existing ltd company is bought, how can new owners check the company's history and ensure there is no liability taken over? Thanks in advance