If the company don’t supply a defect book or checklist then I can’t see how it is the driver’s responsibility to provide one at their own expense, as long as you show the time on your tacho I would expect to be in the clear and for DVSA to take that up with the operator, though I’ve got no experience of this happening.
As for what gets ticked, as an agency driver I’m always alert for defect books where someone has been ticking the 5th wheel tickbox when the vehicle is a rigid 
Fortunately rare, but I’ve seen it several times up to this point