Is there a way to link those sorts of data from weather services to a cockpit selectable screen?
Of course, with technology, everything is possible. As far I as know, for the moment, this system is not implemented on intercontinental airliners.
I understand weather can be displayed on the flydeck
over USA only for the time being (wonder if many airlines invested in the system ?).
Concerning temperature, you have to monitor SAT at all times (especially in this area) not only for aircraft performance but also for icing : it is not uncommon to see the SAT rise well above -40° C at cruising flight levels.
If you get into clouds ->
NAI ON even at FL 350...