It could be a number of issues:
Both issues could be unrelated too.
I'll put down the routes to check in order of logic and price/cost
Most obvious thing to check regarding the "stop" is the security of the connections to your battery, this could have a bearing on the fuse too. They are not the tightest and best connections in the world on an Alfa. Pay attention to the positive terminal beneath the maxi-fuse holder... Difficult to get at and often neglected because of this.
Check your earthing points to the left inner wing area... are they secure and in clean condition?
A relay sticking on can cause surge so try a second hand relay for now to see if that cures it.
While yo have the relay out, remove them all and keep them in order give all the connections a thorough clean up the re-install them.
.... remove the bottom steering cowling and check the connection to the immobiliser transponder ring. It takes it signal from the little block in your key fob but if this connection works loose, it can cut the car out due to loss of signal, although usually a little key warning would normally show on the central panel between the dials, you may have missed it
Check/ remove/ clean / re-fit your connector block to the alternator (battery disconnected first to prevent residual charge) If it's still playing up..... Then try the following.
Get a "free"
battery and alternator check done, many tyre and exhaust centres do this service.
Pay attention to the charge rate.... Does it fluctuate a lot? If so then the alternator may have a faulty regulator thus surge and drop. Many auto electricians can replace just the regulator at the fraction of cost of a new alternator so worth trying them before buying a new alternator complete.
Cutting out or refusal to start when hot can usually be a sign that the crank position sensor is failing... check/ remove /clean/ re-fit the connection block to see if that cures it first.
Good luck mate