Thanks for your question I am Sam and I am one of the UK tax experts here on Just Answer.
Its unlikley that you will need to complete self assessment each year but HMRC would have advised if this was the case - have they advised this is so, or issued you with a notice to complete and fie self assessment ?
(Of course if your income is in excess of £42475 a year then or you have other sources of income, then self assessment is then likely to be the case)
And whilst you state you are not an employee you are being treated as an employee for tax purposes, so then the answer regarding the P11d is yes, as you are treated as and employee and this company reimburse you out of pocket expenses, which will need reporting to HMRC and this is through the completion of a P11d for you.
Of course if they have a dispensation in place, which recognises, that these specific expenses for you are wholly and exclusively in the performance of your duties, and HMRC have agreed through the dispensation that no P11d is required then this company would have advised you, that this is the case, but you do not indicate this.
If your employer has no dispensation then a P11d must be completed by them, and if all the expenses are allowable expenses due to the fact that were incurred through the performance of your duties, then you can make a S336 claim, which asks HMRC to recognise, that all the expenses were wholly and exclusively in the performance of your duties and you should not be taxed on the amounts paid.
This prevents an incorrect tax bill from HMRC!