Hello and welcome to Justanswer!
Let's see if any fault codes are stored. With the doors closed cycle the key from off to on three times ending with it on and watch the digital odometer display. The codes start with a P and have four numbers, there may be more than one stored. Write down any that you see and post them here.
Has anything been done to the intake heater relays?
P0480 is for the electronically controlled fan clutch on the engine. This could be a problem with the wiring to the clutch or an internal problem. Make sure the wiring running to it isn't damaged.
P0541 is for the number 1 intake heater relay. This could be the solenoid itself or a lost power feed to it. This could be contributing to your hard start but won't be the whole problem. These trucks will start and run in very cold temperatures without the intake heaters working.
P2607 is for the number 2 intake heater relay. These are going to take some electrical diagnostics to figure out, which I can help you with if you have a voltmeter.
Your hard start sounds like possible injector wear or leakage. If you are still having problems starting it when the weather is warmer the problem is deeper than just the intake heaters not working. The pressure sensor on the fuel rail has to see a few thousand psi in the fuel rail before it will turn the injectors on. If the injectors are worn or leaking, they will allow fuel to leak into the combustion chambers or the return channel in the head while you are cranking, and the fuel cannot pressurize fast enough. When this happens the injectors don't turn on and the engine just cranks.
The connector tubes from the fuel lines to the injectors can leak. They can be pulled and checked on the end for erosion, and if ok, put back in and retorqued to see if the problem still happens.
To check for wear and injector leakage and wear, a test needs to be done with special test fittings and a scan tool. The return fuel is measured that comes off the injectors-if the amount is too high it means the injectors are worn. If a nozzle is stuck there is a special cap that goes on the fuel rail to cap each line one at a time to isolate the stuck nozzle, when the truck starts normally, you've found the bad one. The high pressure pump output and the limit valve will need to be tested. A leaking limit valve will allow fuel to return back to the tank when you crank the engine, and obviously low output from the high pressure pump won't create enough pressure to start the engine. Low output from the high pressure pump will normally set a P0148 (fuel delivery error) code though. All of these tests have to be done with special equipment that a dealer or Cummins shop will have. There are special fittings and a scan tool needed .
Remember to never open a high pressure fuel system component while the engine is running or cranking. The possible 30,000 psi can penetrate your skin.