John - FR1 and DL are the application forms for a first registration. You don't need form AP1
You can only ever transfer the whole legal ownership so the two of them would be transferring the whole to you and your Mother using form TR1
Our online guidance re transferring ownership and first registration explain what's required - the registration fee will be Scale 1 on the current market value of the property
Practice guide 1: first registrations - GOV.UK and
Registering land or property with HM Land Registry: Change the registered owner name - GOV.UK
In light of your scenario, the Qs asked and the interest/TIC aspect I would strongly recommend you seek legal advice/assistance also. It's important to consider the legal and financial aspects of such matters and what options are available and what's 'best' for all with regards not only the now but also what may happen in the future