Statutory work is providing the psychological advice for EHCPs. It varies a bit, but mostly it involves information gathering from school, the child themselves, and their family, and then writing that up into a report summarising their strengths and needs, and writing outcomes and provision. The info gathering can take lots of forms, e.g. a meeting, observing the child, 1-1 work with them (and all the different possibilities that involves), joint meetings with school/parents/other professionals, etc. The reports are often quite long and time consuming (though I know in my LA we're working to try and reduce this), and it can feel a bit like we're being asked to gatekeep resources, and firefight rather than coming in early before things get to a stage of needing an EHCP, which is frustrating. But then again, as one of my tutors said, it pays the mortgage!
For your other questions - I use my own car, and I think outside of a city centre authority it's nearly unavoidable. I don't get paid overtime, though I try to minimise how much I work beyond my hours, but it's sometimes unavoidable - it usually evens out throughout the year with some less busy weeks in the holidays. Holidays are often for catching up on admin/report writing, plus some nursery school work in ones which open all year, but noticeably less frantic than term time, where it can be a bit of a rush to fit everything in. Lunch break-wise, I plan my own diary so always try to put time in to stop, usually half an hour or so - sometimes less, sometimes more if I'm with colleagues and want a catch up! And I'd usually bring food from home if I'm out, but am not always organised enough so might pop to the shop or a cafe.
The job is very varied, so I wouldn't say I have a typical week as such. The type of things I'd be doing could be meeting or observing children, having consultations with school or parents, delivering training to school staff, supervising other professionals like ELSAs or speech and language therapists, as well as planning all of those things in advance, and report writing and admin (and I'm sure other things which I've not thought of!). I work 3 days a week and would usually try and work from home once for admin.