Whilst there is no official definition of Flexi-schooling, it is widely accepted to be an arrangement where a child attends school on some days and where the rest of the education happens off-site (at home, at wood school etc).  It is still classed as full-time (FT) education.

Yes. It is still currently legal to enter into a Flexi-schooling agreement, though the likelihood of this being an option seems to be reducing with the national issue of schools already being oversubscribed with full-time pupils.

Schools are not obliged to mark pupils as absent. Code B or C could be used. Nowhere in the law does it say approved education off site involves the school being responsible for the welfare. And it is an untenable demand; not in statute and not following case law where a child suffered grievous harm at a swimming lesson and the school / LA were taken to court, the court quite rightly said so long as the school had made reasonable steps to make sure the lesson was safe (it was a public baths with trained lifeguards etc) then they could not be held responsible.

No. The decision is entirely in the school’s hands.  Many schools aren’t aware of FS, that they get full funding even for FS pupils, and that FS can work to the advantage of both the pupil and the school.  But if you can’t convince the head – or the school already has a policy to not accept FS pupils – your only options are FT school or FT home education.  Note: for under fives, full-time school is not a legal requirement, but the head may require FT if it’s an oversubscribed school and you wish them to hold the child’s place.

Official DfE guidelines to LAs on FS & Attendance