Do I need to keep a logbook?

My monthly salary is split up into salary and travel allowance. Therefore my PAYE is calculated accordingly. The company pays for all my petrol, whether business or private. Do I still keep a logbook and can I still submit a claim?

Yes you can keep a logbook, but you would not enter a value for petrol then when you complete your tax return.

Would I be able to submit a claim against other running costs when I do my tax return?

Yes, insurance and maintenance etc will be allowed if you pay for them.