First task is probably to download the probate form and associated notes. You also need to fill in an IHT form - there is one specific to not paying IHT if I remember correctly. Look on .gov site, am assuming link above will help.
The IHT form needs estimates of the estate, which will include cash and the house. You can get 3 valuations from EA and just go with average. Assuming you are selling the house, you will need them anyway. It will only be a problem if you are very close to IHT levels and sell for more than you specify.
The form will also ask for estimates of house furniture etc, when my mum died I just put a ball park figure down as most was going to charity shop, with just odds and sods sold.
Who have the insurance policies paid out to - all the cheques I got from insurance were paid to me as the main executor? I am not sure if they are part of estate, I think so?
My DB and I were joint executors, although I did most of work. To save us having to jointly open an executor account, we just used a dormant account of mine, both signing the letter to the banks to transfer monies across. It sounds like an executor account might be needed in your case though, I dont think you will need probate to open one, but maybe to access it you will.
Have you got proceeds of bank accounts even without probate? It depends on the amounts I think, so you may have.
In terms of funeral bill, mums bank made a cheque out to funeral director from her account, whilst it was frozen, it is one of the few transactions that can be done whilst account is frozen prior to probate.
Everything else, I paid for and claimed back. Make sure detailed accounts are kept - just a spreadsheet is sufficient but maybe separate tab for credits, debits, amounts paid out by your DP etc. Keep receipts for latter too.
Hope this helps, probate is relatively straightforward, esp if under IHT limit. Your DP will need to sign an oath, that can be done at a probate office or a solicitors. I think from application to getting certificate was about 3 weeks for us.