I would not think that the fuel pump should run with the engine not running. The usual sequence is that the pump comes on for a second and then shuts off if the motor isn't running 300rpm or if the starter isn't engaged.
Even though you are from a warm region (I am from Florida), the temp of a cold engine is still very different from a hot one. The cars system will work well in all temperatures when performing properly. I do not know what system you have as your year is very close to some major changes. I am not even sure what engine that car has. I presume it is a 126 body 117 engine with a KE-Jet fuel system.
Read the DIY section about evaluating engine controls and find someone in your area that understands the testing, or get the tools and give it a go. (all the tests can be done with a $30 multimeter, a LED, and some wiring)
Bosch Master, ASE Master, L1
33 years MB technician