The gear selector is on the top of the gearbox assembly with two cables running to it... A local specialist Alfa garage can make short work of it in less than an hour and have it aligned to perfection mate. It's tricky to DIY if you aren't too sure as its by feel, trail and error really.
Battery tray needs to be removed to get the room to do it properly so you can imagine that takes a lot of time up fitting and refitting when going the DIY route. If after inspecting the rest below doesn't do the trick... Then look into the selector
I would hazard a guess that the engine / gearbox mounts are more to do with it jumping out of gear. Especially if the car has been standing a while idle... Rubber likes to move in order to keep it pliable, when it's sedentary in varied temperature, that's when it weakens and cracks. A good push and pull with pry bars can usually reveal any weakness.
A new fuel filter is a MUST. I can't emphasize enough how regular changes of a fuel filter on a JTD can do wonders at preventing many common issues... For the cost and ease change it every 6,000 regardless, especially after EGR/ MAP cleans and miraculously these problems rarely occur again!
Check your induction pipe work too, anything that is rubber for cracks and hairline splits that can be bleeding air (refer to the properties of rubber standing for long periods).
Remove them, seal them up with inspection gloves and dunk test them in a bucket of water to check for air leaks...
Once all that is checked, take the car out down a motorway if you can and get the car in third/ fourth gear at a steady 70mph with higher revs than normal and leave it constant like that for 15 miles or so so any crud that has settled in the fuel/ induction and exhaust system get a gentle constant burn out at a higher and steady temperature.... Chances are it will do the fuel system/engine and cat the world of good.