Why does the Machine call Finch on phone booths, when it can contact Root on a large range of devices, including mobile phones?
Finch designed the Machine to never contact anyone other than the government (with relevant/terrorist persons-of-interest) and Finch (with irrelevant persons-of-interest). Part of the Machine's design was a limitation on what information was allowed to be passed (the SSN of the person of interest).
I have only watched the first three seasons of the "Person of Interest" series so far, and from what I remember of those episodes there was no in-show (canon explanation) of why the machine uses public phone booths to contact Finch, but I have a theory as to why this is so. Be aware that future episodes of the show may provide a more authoritative answer to your question.
Finch designed the Machine to only use public telephones to protect himself from being located by the government if it ever discovered and traced those calls. The Machine's code also included limitations on who the Machine was allowed to contact and a filter on what kind of information it was allowed to pass along (only the coded SSN of the person of interest).
At some point the Machine created its own code to get around the limitations of its original design/programming code. The new code removes the limit on who the Machine can contact (allowing the calls to Root), removes the limitation of only using public telephones, and removes the "SSN number only" information filter. This last is shown in episodes where Root (and sometimes others) are given detailed instructions and information about their surroundings by the Machine.
The Machine continues to use the original code while passing SSN information to either the government or Finch because Finch designed that code so that the machine could not modify or delete it. When the Machine wants to talk to Root or anyone other than Finch or the Government it uses the new "unrestricted" contact code, which also allows it to choose what information to pass on and what method of contact it uses.