They say you have to keep papers for SIX years however, any liability to the Revenue is not statute-barred.
To register a trademark is not a cheap exercise. The registration fee is around £200 plus a fee per classification. You may need advice from a Trademark Attorney and the one I can recommend is Philip at
http://www.crossguard.info
Going back to the paperwork there is less paperwork for a sole trader but some forms and accounts have to be submitted. You can earn I think it is around £4800 per tax year without any liability for income tax. You will still have to register as self-employed with HMRC and you must do this within 3 months of trading to avoid any fines.
You mention that your mother is self-employed. I presume she has an accountant on board? You could obtain advice from the same accountant. If she is not registered and earning money then she also risks being fined
Jonathan