I race karts regularly. I have first hand experience.
I'm assuming that you are talking about short-circuit karting (i.e. non-gearbox). Long-circuit karting is a whole different story (and one where I have no experience).
If possible, try your boots while sat in the kart. I have some really nice, comfortable Puma racing boots but as soon as I get in the kart I find that the rear portion of the ankle cutaway simply rubs/presses on my achilles heel - not especially comfortable after even 10 minutes of racing (and I usually race for 90 minutes - these boots are now gathering dust!).
In terms of high-top versus low top, there is a marginal argument on the grounds of safety and protection (imo) but in reality I think this is negligible in karting and it simply becomes a matter of comfort and protection. Similarly, space restriction is rarely an issue in a kart.
In the UK at least (and I'm pretty sure any FIK affiliated environment), flame/fire retardancy is not a requirement at all given the low-risk of fire. You will be pushed to find kart-specific equipment that is fire retardant. The consequence is that karting kit generally looks and feels the same as circuit racing kit but is considerably cheaper.
As far as fit is concerned, comfort is key again. Kart boots have thin soles to assist in the "feel" of the pedals but unless you have a gearbox kart, you won't be swapping pedals with your feet so "sensitivity" is much less important than in car racing.
I meant to add - the presence or not of a cut out for the heel will very much be a factor of your (son's) stature and consequential position within the kart. A taller driver will need less "pivot" in their heels and will approach the pedals from a more "top-down" position - meaning less room required in the back of their boot. A shorter driver, like me, will have their legs a full extension (almost flat) and therefore their ankle more pivoted and require more room at the back of their boots. Again though, it's comfort!