The shopping and cleaning are easy answers. Whether you phrase it as such or not, by providing a service for money you are in business. That is all you need to do at its simplest to start a business - you don't have to register for VAT exemption until you are turning over about �50,000 and that's a lot of cleaning.
If you are in receipt of any benefits, working for any other pay has implications so you need to keep careful records of incoming and outgoing monies, so DSS can see if they want. These records will also be useful for showing Inland Revenue if asked. You ought to do a self assessed tax return once a year, but if the figures are very low it should not be difficult.
So get hold of any of the very useful books down the library about small business ventures. The other stuff, showering etc, be very cautious about as you will be wide open to accusations of impropriety.
It may be, if you make contact with your local agencies like Help the Aged and the Council, that they want someone to do these things and are willing to pay a small wage and provide training, especially in the safeguarding aspects of working with the elderly.
There may also be local organisations that can help and advise with managing your enterprise.
If you go ahead on your own, you will need to consider insurance.