Doubt you have a problem on any of your questions. If you made a gain on the disposal of your mother’s property (compared to the probate value) it might breach your capital gains allowance (£11,000?). That’s probably unlikely if you are selling straight away.
If you have a few £ now, I think it would be money well spent to have a chat with an accountant. Get a recommendation from someone you trust. They don’t cost the earth and keep you the right side of the law and ensure you don’t pay too much tax.
Edit: your inheritance shouldn’t be taxed and I’m not aware of it needing to be reported.
HMRC do have access to bank records and the best advice is always to keep records of incomings and outgoings to show the source and use of funds. It’s particularly important if you don’t run a separate business bank account and are in a cash type of business.