In regards to your questions:
1. I'm not sure if it needs mentioning or not, since companies move offices/change logos all the time - this isn't a question I've ever been asked before

.. I wouldn't have thought it's anything to worry about - concentrate on the important stuff (making sure your bank statements/payslips and employer letter meet the requirements).
2. Don't think it'll matter. Again, signatures can change.
3. I don't think anyone will even notice this. Remember, the contract is not a required document, it's just recommended.
4. Nope. Most people just have someone in HR write their letter! (Plus, as mentioned, the contract isn't a required document anyway)
5. Easier just to write one and post it... though, really, it should be your employer that writes it, not you! You can draft one though, so you know what needs to be included and can make sure your company includes everything required
6. You must include the FULL bank statement, so send all of April's statement.
If you're applying in October, you will need to send the following payslips (assuming you are paid at the end of each month):
March 2018
April 2018
May 2018
June 2018
July 2018
August 2018
September 2018
And the following bank statements:
10th Mar - 9th Apr
10th Apr - 9th May
10th May - 9th June
10th June - 9th July
10th July - 9th Aug
10th Aug - 9th Sept
10th Sept - 9th Oct
7. No, if the payslips are digital, they must be accompanied by:
"a letter from the employer on the employer’s headed paper and signed by a senior official, confirming the payslips are authentic"
This can be incorporated into the main employer letter, and it should state the amount and date of every payslip