We recently had a Cooker Hood installed, a Hotpoint HE63X. This is in
filter mode, there is no vent to the outside.

However, there is only the two fat/grease filters, no evidence of a
carbon filter. The instructions state the carbon filter must be
provided if there is no suction.

I don't see the point of this hood in it's current form, would it be
normal for the kitchen supplier to provide the filter? Also, there is
nowhere for the air to leave the hood other than back out the way it
came in, is this right? The pictures show a vent in the chimney, but
there isn't one. I would think it dangerous to blow air back onto a
cooker.

I don't think it's much more than decoration! I don't know if the
above is normal, Hotpoint's instructions are not very detailed, but
the kitchen fitters have got nothing else right, so is this another?