'Is it possible to do a trip to Granada from Málaga and see the Alhambra in a single day using public transit?'
Absolutely, many do. The bus company Malaga to Granada is www.alsa.es and I suggest a taxi to the Alhambra from the Granada bus station.
'Will I still need to book a ticket in advance to visit the Alhambra?'
Certainly book as soon as you know when you wish to go, there is a limitation on numbers per day (possibly not relevant in November), you will avoid time expended queueing and have peace of mind and a time slot for the Nasrid to suit you.
A guide - http://www.tripadvisor.com/Travel-g187441-c77529/Granada:Spain:Buying.Alhambra.Tickets.Online.html
Malaga city is your public transport hub, a normal working Andalucian city with it's own sites to see. It is not as touristy as the Costa del Sol, but has become more so in the last couple of years due to port expansion and more cruise ships.
Sorry, no recommendations for accommodation from me, except I am happy with the IBIS Centro a couple of nights a year. Plenty of choice on directories like booking.com with testimonials from those who have certainly stayed at a premises.
Have I covered all?