Sounds like one of two possibilities:
1) Loose connection, since it works intermittently after you push/pull/change something
2) A problem with the main control board. Everything from the cooling system to the lights to the fans to the ice maker are managed and controlled by the control board. Some boards have flashing LED indicators to help trouble shoot problems.
Usually, the beeping will mean that the fridge thinks the door is open. You did not give a model # so I do not know if it is a side x side, top freezer or bottom freezer.
The evaporator fan is wired through the light switch so if the door is open, or there is a problem with the switch, then the fan will not circulate air and typically things will start to freeze toward the bottom but be warmer at the top.
The first thing I would check is to make sure that the top door hinge covers are securely in place and that the switches close when the door does. Sometimes it is a mechanical switch while others use a magnet. Look for something like this on the top of the fridge just on top of the hinge. Post more details if more assistance is needed.
The effort it would take to eliminate filter would be too much. Just leave filter in place. If new house filter is working right you should never have to replace fridge filter and your water will be double-filtered :)
Check to see if the fridge and the freezer are overfilled Over-stuffing in the freezer or the fridge or both of these compartments will prevent the doors from closing properly and which can cause the fridge's to set off the beeping alarm.
Check the fridge door seals: Broken or damaged door seals may stop the fridge from closing all the way. This can also happen if the seals are dirty. Wipe clean the door seals with a soft cloth and warm and use soapy water and close the fridge doors. If the door still appears ajar then you have to replace the door seals.It is also quite possible that the inbuilt door-alarm which sets off if the fridge door is ajar for more than a few minutes has gone bad . This would need to be looked at by a qualified repair person . Until, you can call a repair person transfer all the food items to another cool place.
Check the fridge balance: To work correctly the fridge must be placed on a levelled surface. Even a slight tilt in the fridge structure can set off the beeping alarm. Check to see if the fridge is balanced evenly.
Check to see if the fridge exterior is warmer than usual: Check if there is adequate distance between the fridge and walls to allow for proper ventilation. If you have already ensured this and the fridge exterior still appears warmer than usual, have an appliance engineer to take look at the problem. If the beeping continues, switch-off the fridge.
Check the freezer: If your fridge does not have an auto-defrost function then excessive ice-build up in the freezer can cause the beeping alarm to go off . If you see a lot of ice build-up in the icebox or freezer, switch off the fridge and let the ice thaw. Once the fridge is thawed, the beeping should stop.
Have a repair person check the circuit board: It is possible that there is an issue with the appliance's circuit board. If so it will most likely need replacement.