Can my extractor ducting go into my chimneybreast?

Be careful with chimneys as it can have all sorts of nasties in it plus your ducting should run unbroken yo the outside were possible have you thought about a recirculating cooker hood

Yes find a cooker hood with a good carbon fibre filter ( recirculating ) if this is changed and cleaned regularly if will do the job required it’s common to get properties where ducting to outside is impossible good filters will be good

I only use the Chimney for open Fires and gas fires. The best way move your cooker or you can lower your ceiling and you can buy a flat ductwork to going to the ceiling and out another wall.

If I do not remember wrong your are allowed only 2 off 90 degrees bent in 4 meters pipe line and the last bit that goes outside should be not in level but a bit off set towards the exit, to avoid condensation to come back into your pipe line (I am talking for a wall hole to go out). As far as I remember you should not connect your fan to the chimney. I mean you should invite the air to go out so if you can break into the chimney and fit an elbow that goes up it should be OK. If you go straight in into your chimney all the small can go down and no up, depending of pressure. Specially in winter an air block in the chimney will not allow your extractor to work properly. Hope this helps.