Being paid through PAYE means income tax and national insurance is deducted at source, i.e. deducted by your employer and shown on your payslip - the amounts you pay depend on PAYE Allowance & National Insurance Thresholds. I'd recommend finding out more about these on our Crunch Knowledge website. As a Sole Trader, your earnings are reported on Self Assessments every year where you'd pay personal tax to HMRC instead.
Spot on! Calculate the cost as a total and claim the correct mileage allowance. For vehicles, the mileage rate for 2022 is 45p per mile for the first 10,000 miles and 25p per mile after that for business-related driving.
This should be allowable, as long as the trip is purely for business purposes. If there's a personal element to it, this may be seen as "duality of purpose" and could be rejected as an expense claim by HMRC.