2010 Chrysler 300 Speedometer not working

You have an intermittent problem. Fuses are never intermittent. When they blow, they're done, and that circuit will never work again until it is replaced. There's no relay's involved with the speedometer. Relays are used to power up high-current circuits. The most likely suspect for an intermittent speedometer is a failing speed sensor on the transmission. The cruise control system needs that information too, and that would explain why that stops working at the same time, but it's hard to know if or how the other symptoms figure in. On a lot of newer cars we're running into things that stop working due to causes that are seemingly totally unrelated. I would concentrate on the speedometer first, then see if the heater problem is still there.

The place to start is by reading the diagnostic fault codes. Chrysler makes doing that yourself very easy. Cycle the ignition switch from "off" to "run" three times within five seconds without cranking the engine, leave it in "run", then watch the code numbers appear in the odometer display. Reply back with those numbers, then I'll see if they could be related to this problem.

P0121 - Throttle/Pedal Position Sensor/Switch A Circuit Range/Performance Problem
P1110 - Atmospheric Pressure Sensor Low Input
P1120 - TPS Out Of Range Low

At first glance, none of these have anything to do with the speedometer. We've been running into a lot of weird symptoms though that seem to have nothing in common with their causes. A friend recently ran into a truck that had a very noisy transmission, and that cleared up after repairing a broken wire for the ambient air temperature sensor. That one really surprised us, but we've been seeing more and more things like that the last few years.

You didn't list the right engine size so I'm not sure which throttle system you have, but if it's the "throttle-by-wire" system which adds another unreliable computer to the car, I can see where these things can all be related. There's two things to consider. First, have the dealer erase the fault codes, then test the electronic throttle system. If that solves the problem, at least for now, they may be using the speedometer as a way of getting your attention so you have the throttle system repaired. In my opinion, electronic throttle systems are dangerous and I'll never own a car with one. That's what put Toyota in the news a few years ago.

The second thing the mechanic can do is use the scanner to view live data to see why the cruise control won't set, and its "reason for last cutout". I suspect the "reason" will be "vehicle speed too low" or "below minimum threshold", but that will just verify what we already know about the missing speed signal. From there, he can see if multiple computers aren't getting the speed signal, or if just one computer isn't reading the signal being sent to it.

P0645 - A/C Clutch Relay Circuit

Please start a new question specific to your vehicle. This was a private conversation between two people. As such, none of the other experts are going to see your addition or have a chance to reply. That won't get you the help you need.

You can find fault code definitions all over the place. If you want help solving a problem, please include the symptoms and observations as to what kinds of things affect it, and what you've done already, and the results.

Please be sure to list the engine size, transmission type, and mileage. We use that information when making generalizations as to best suspects and which tests to perform.
Here's the link to take you directly to where to post your new question:

https://www.2carpros.com/questions/new

You can find the generic list of fault codes here:

https://www.2carpros.com/trouble_codes/obd2/p0600

Some fault codes are manufacturer-specific, so they won't be found on this list.