I think you have to declare the income and prove the expenditure. You can do two jobs but must pay tax on all your income. If you cant prove the expenditure then the HMRC will treat it as earned income and tax it accordingly
Yes- and assuming your main job pays more than the personal allowance (around £8000pa) you need to pay tax at 20% on all your earnings. I'm not sure about NI though- I think that may apply separately to each job so you may fall under the threshold for NI on mystery shopping earnings.
Yes, they will just send you a self assessment form. You fill in the PAYE section with the info from your P60 and provide them with the info for any extra work. Keep ALL receipts.