Is the Bryant 661CJ030-A heat pump single- or multi-stage?

This model can use a generic contactor providing it is rated for the amp load of the compressor and fan of the unit. I will link a similar contactor that you can use.

https://www.supplyhouse.com/Trane-CTR2573-24V-2-Pole-Contactor-30-AMP

heat contactor stuck in.

if you have 240 volts at contactor you either have defective contactor or the 24 volts is not there.to pull contactor in.

Your t-stat should be set up so that the furnace is the second stage heat and also emergency heat. First stage (heat Pump comes on) sometimes they use Time to start 2nd stage, sometimes they use outdoor temp and this all bepends how bryant has set it up, without a model # I cannot tell you but send it and i will look in somebooks i have to tell you.

This should help you. You need to know if your thermostat is powered by the unit or independently and which way your reversing valve switches (I,m not sure on a Bryant, but probably Your old O wire goes to your new O terminal. They should pretty much match up, the new stat just has more options. Thermostat wiring

Refrigerants are regulated by the EPA. In order to purchase it, you need an EPA License. If you are low on refrigerant, the leak needs to be found and repaired. This requires special tools.

Additionally, with the information given, it could be an electrical problem. Is the outside fan running, is the compressor running? Your best bet is to call a professional.

YES. It seems your capacitor is bad.
It must be replaced with the exact specifications as the original.
IE: MFD and VOLTAGE for the compressor and fan motor.
Example...35+5 370 vac.

The 44360 tstat can be used for single stage Heat pump use only. Unless you are in an environment that has greater than 40 degF outdoor temperatures year-round, you will want to return that tstat and obtain a multi-stage Heat Pump tstat. The standard system is 2-stage......1st stage is Heat Pump Compressor and 2nd stage is the backup electric heat.

Hope this helps. Good Luck. :-)
http://www.hunterfan.com/uploadedFiles/Support/Downloads/42710.pdf?n=149

This may be as simple switching the thermostat to electric, under the cover is a gas/electric dip switch. Hope that helps.

Found charge capacitor top ballooned replaced with a universal capacitor found at a local appliance repair shop system works like a champ capacitor is located behind the interior electrical panel after removing the side panel