SpotTheCat wrote:Won't start (though the starter solenoid clicks)
If the starter solenoid on the starter motor itself is clicking and not turning the engine over then either the starter motor or the wiring between the starter motor and the battery or battery and chassis is borked. I suppose the engine could be seized but that is very unlikely.
Commonest one is the starter solenoid is sticky and a bit of percussive maintenance may get it to fire up. Given new battery it may be that the connections are not good (someone didn't put them back on properly), this would also explain the key fob problem. I would first check over the connections I mentioned looking for loose ones and probe them with a meter looking for volts across the connection whilst someone else turns the key.