welcome to the world of Alfa, and to the forum.
sounds like you'll be spending some time on here, but panic not, youre in good hands!
firstly the air con. dont think i'm taking the p155, but firstly have a careful read of the owners manual regarding instruction for the digital climate control, it is a hellishly complicated system unless you really know it, and yes it would soon appear to have a mind of its own in one of its automatic settings! mine still confuses me to this day from time to time!
when you want cool air con, are you hitting the snowflake button first, then trying to turn the temp down?
best way to engage setting for a cool cabin is have it set to "mono", then using the heat control on the drivers side (which will control all zones if mono selected") turn it all the way down until it go below "16.5" and becomes "lo" on the readout. This should bring in the air con snowflake light too, then use the central control to adjust the fan speed and the direction.
if this fails to work, then you may have an issue, but make sure the car is at some sort of working temp before you try.
speaker certainly sounds like a poor connection, but it could be at either end. easisest to check the head unit end first, they are all in a multi connection plug, unless of course theyve been butchered for an aftermarket unit fitment?
as for the info display, its quite common, but its not a totally dead screen.
if you get you air con to blow cold, send it up through the top vents to the screen and to the face vents, this will in turn cool down the infocentre display, and you will often find that the contacts that have expanded with heat will close up enough to improve the display.
the only permanent fix is to dismantle and reconnect the screen display feed in a tighter position if you get my meaning. Ive been intending to do my own but as i'm using the air con a lot at the mo, its not bothering me half as much as its been pretty good to look at once colled down a bit.
see how you go with that and let us know