with regards to your key fob problem, it may be due to strong ambient interfearing signals overriding the weaker radio frequency signals from the fob, this diagnoses can be ruled out by taking your car to some other distant place and see if it does it elsewhere, as it does outside your home or in the car park you mentioned, if it happens in most places then the issue is likely to be the fob itself, strangely how silly the Nissan dealers changed your car battery instead of changing the key fob battery as a first port of call, I would try and complaint about that and get your money back or request your original battery back and return the new one, that you need not have spent your money unnecessarily on a new battery. Try changing the battery in your fob as that may have gone weak.
On another note, do you fumble with the key fob like having a habbit of pressing the button whilst in your pocket well away from the car, because if you do that, then there may be a possibility that the codes emitted by the key fob get unsyncronised after a few presses of the button and therefore requires a re-sync by pressing the button a few times for a few seconds to sync again. Have a read in your car's hand book. This is a feature in some key fobs using so called Code hopping techniqe where the codes cannot be grabbed as each time the fob sends a different code and the central locking system receiver knows the next few codes it is likely to receive and if it receives a code outside the allowed limits then it won't open the doors. Do you get the same problem when locking?
With regards to your stereo, check all the power and speaker connecters at the rear of the radio, and possibly a loose fit fuse.