I am planning on setting up a private limited company with my brother. We are going to register with the companies house with the business name (example) "123". Once we are registered, and have a business bank account, will people be able to write cheques out to be made payable to "123"?
A UK business bank account registered in the name of a limited company (e.g. 123 Ltd) will always be able to receive cheques made out in that name.
Most banks will also, subject to receiving proper notification and completing some simple forms, allow you to accept cheques made out to a trading name.
So for instance, if you incorporate as 123 Ltd, but operate "Trading as Abc" (which is often abbreviated to "T/A ABC"), then as long as you have cleared this with the bank in the first instance, you will be able to accept cheques made out in either name. Official documents (such as invoices) must always include the actual business name.
Yes. When you open your business bank account, you will need to show them that you properly registered the business. I don't know all the details in the UK, but it should be the same as any other country. The business account name will have the name of the business on it and cheques made out to that business will be depositable into that account.