EU child passport application, supporting documents.

[Heyley]

Hi,
I am in the process of applying for a British Passport for my child based on my residency status in the UK at the time of his birth and I am looking for some help please.
My child was born in August 2016 and my timeline is as follows:
15. July 2011 - 1. September 2011 I was a self employed intern.
At the end of my internship I was offered a job and worked for the company from 20. September 2011 to 1 August 2014 when I was made redundant.
I was out of the UK for a couple of months and registered as a jobseeker when I returned to the UK at the end of September 2014.
I stopped being a jobseeker when I found a new job in December 2014. I still work for this employer.
To prove 5 years continuous residence I will need to provide evidence for the period between August 2011 - August 2016 so I am looking for some guidance here.
First of all, my biggest worry is proving my self employment for that one month in 2011. I do not have any paperwork although I was registered as self employed and can show a bank statement which proves that I was paid during my internship. I also paid NI contributions on this amount. However, when I log into my Government gateway account I can only print out an overview of the tax I paid, it does not go into any more detail with regards to the dates of the self employment etc. I cannot access my SAR02 for that period also as that was more than 4 years ago. I also do not have any communication relating to my self employment as that was all done through the company's email which I no longer have access to.
Second, I am also missing a couple of P60s from my employment.
Therefore, my questions are as follows:
If I ask HMRC for employment history, will that include self employment? If not, is there a way to obtain proof of self employment?
According to the guidelines for passport applications for children born to EU parents, an acceptable proof of residency for workers is also a NI contributions statement from from HMRC. I have 5 full years of NI payments, would it be sufficient to obtain proof of this for 2011 - 2017 and supply it without going into any further detail with regards to my employment and self employment details?
And finally, do I need to supply proof of my absence from the UK in 2014? I do not have any tickets, as I traveled by car, so what can be provided as evidence if this is required?
Thank you very much in advance.