Which parts of the Alhambra in Granada are free?

The Charles V Palace is free. If you go through the Justice Gate entry, the grounds in front of the Nasrid palace entry (bw Nasrid entry, Charles V and the Fort) are all free. There is also an Arab bath and a Christian church around this area, free. Then there is the Parador hotel and gardens around it. Free, and you can go eat or have a drink at the Parador restaurant. Basically, tickets are checked at entry to specific points (Nasrid, Alcazaba, Generalife and palace etc) but most areas between the key points are free.

'Which gate would one go in to walk around "free"? ' - Any gate. The ticketed areas are individually entranced. If walking up the Puerta de la Justicia is about a couple hundred metres short of the main gate.
Ticketed areas are the Nasrid Palace, the Alcazaba, and the Generalife. Monday will find at least the free museums closed, if I recall correctly.

Thank you so much. That is good to know.

Is the complex open and free in the evening also, so that even if you don't buy a ticket for the palace or the Generalife Gardens, you could still enjoy the lighting of the Alhambra?

Yep. I went to the Nasrid Palaces evening visit and walked through the entrance. They only looked at my ticket at the entrance to the Nasrid complex itself. There are also evening visits to the Generalife, so that part of the complex would be lit enough for safe walking. I doubt any of the other buildings would be open.

By the comment do you mean the Charles V "museum", or other museums in Granada? I haven't seen any other reference to sites at the Alhambra being closed on Mondays and that is probably the day we will be there. It can be adjusted to Tuesday if necessary, but would not be easy to do! Any information on closures at the Alhambra for Mondays?

Oops...was referring to Neil's comment about Monday closures for the free "museums"!!