Without taxes submitted or current wage slips it will be very difficult. I am self employed, they needed evidence of current earnings for the last 3 months to prove each week I had earned over the £120 a week minimum as my last submitted tax year (2015-2016) I was only working for a small part of it as I gave myself 18 months maternity. You will need to do the same for your partner. Both must evidence earning over £120 a week consistently.
Do you have a National Insurance number? Applying for one will be your first step - but if you lived in the UK unil the age of 16 you should already have one. Call HMRC and ask. You'll then have to register as self-employed, in fact you should probably have done this already. It doesn't matter where your clients are based - it's that you're here and should be paying tax and NI here.
No idea about the benefits thing or free childcare but it will depend on lots of factors, including earnings.