could it be crappy fuel ?
changing the spark plugs couldn't hurt especially if you don't have any maintenance records.

Im not an SME on your chassis, but Winter can affect the car via the fuel. Try some Heet (RED BOTTLE) and see if that helps

 

Have your motor and transmission mounts checked.

 

We checked those, transmission seemed ok, motor mounts are not as great but specialists said they are not the cause in this particular case with such violant jerking as it would have persisted in other scenarios too.

I will have the spark plugs changed to be sure there is no problem there..

Any thoughts on having the injectors cleaned? Car may have been parked for a long time and I was told the heads tend to dry up or something..

 

If the car has been sitting it is certainly worth running a tank of fresh fuel with a bottle of Techron fuel system cleaner. The car may just need to be run for a while. Try a highway trip and once you have fresh fuel do the occasional brisk acceleration - wide open throttle up to a legal highway speed (also known as an Italian tune-up). These won't fix an actual problem but sometimes the fuel system just needs some cleaner and higher flow to resolve an issue like this.

 

I will be using this fuel for the next month and it should clean the system up I am guessing.

Efix Gasoline 95 | Rompetrol, energie vie

As for the spark plugs, changing them tomorrow.

 

I changed the spark plugs and one of the things that goes above it as it was leaking electricity. The car still has the problem after resting in the cold for 30 mins to 1 hour. No problems in the morning when super cold.