Ilona,
I have never heard the first rumour in your post about lodgers having any right to claim any part of the property value, no matter how long they have lived in your house. This could possibly only take effect if they were claimed that they were a live-in partner.
You should always have an agreement to cover yourself against any problems that could crop up, in your case you need an agreement called "House share agreement - resident landlord" which you can either google for (can be got for free) or buy one at a large Stationer like WH Smith (I think a company called Lawpack produce them) then just fill in the names, dates and rent paid.
Always keep records of rent paid and any expenses, you are allowed £4,250 Rent a Room income Tax Free, Anything over this amount is taxable at your own income tax rate and should be reported to HMRC.
You should not need a solicitor for this agreement, but if your mortgage company were to become aware of you renting to lodgers they would then require one of their own forms to be signed by any co-habitants (lodgers, spouse or anyone over age of 18 not currently named on mortgage paperwork)