F4 error

sounds like control board defective, technician will have to verify.

read the mixture instructions
try preheating the oven for the required temperature or when the thermostat turns the light off
insert the cake to be cooked and leave it there for the desired time
Too hot will burn the outside with out cooking all the way through
check the ingredients carefully as if the mixture is too wet ( butter, milk, water all make it wetter than required )
Place the cake in the center of the oven ( not on the bottom shelf or the top shelf)
All these may sound mundane but they all point to getting a good cake result

The Fault Code/s will show up in the appliance Display
F0 - No Alarm Signal
Replace Main Electronic Control
F1 - Door will not Lock
Check door lock. Is Door Lock Getting Power
F2 - Door will not Un-Lock
Check door lock. Is Door Lock Getting Power Supply? Check Control Thermostat
F3 - Self Test Error
Disconnect Appliance from Mains Supply, Wait 5 Minutes and then Reconnect to Mains Supply
F4 - Shorted or Open Circuit Oven Temperature Sensor
Replace Oven Temperature Sensor, Check Sensor Harness Connection between Sensor and Control Unit
F5 - Relay Contacts Failed - Sticking
Replace Main Electronic Control
F6 - Power Board Temperature Too High
Check Correctly Installed, Check Cooling Fan Assembly, Ducting and Airways
F7 - Incorrect Polarity - Mains Supply
Check Mains Connection - Live & Neutral round the wrong way
F8 - Power Supply to Interface Problem
Check Wiring Harness, Replace Main Electronic Control
F9 - Procesor is Auto Resetting
Disconnect Appliance from Mains Supply, Wait 5 Minutes and then Reconnect to Mains Supply

My guess is the element in the back of the oven took that moment to die. You will have to get an appliance engineer or good DIY electrician to check the supply to the element to be sure. While you have it out you can write down the serial numbers etc off the label on the back. You will need them to purchase a new element. Just go online to an appliance spares specialist.

If you are getting soot residue, it could be a number of things, First, is this a natural gas or propane oven? If you bought an oven to be used for propane, it may need to be converted from natural gas to propane or it will soot everything up. 2nd, have you recently spilled food/liquid onto the oven burner? If the burner holes are partially plugged, the remaining holes will burn large yellow flames and soot up the oven. Check the burner flames, they should be mostly blue with little to no yellow tips. Also make sure the oven burner flames are not too big. They should never exceed the width of the flame spreader pan directly above the burner, in fact they should be less than half the width. You could also have other issues like improper air shutter adjustment to the oven burner or very low regulator pressure, but most likely it has just been spilled on recently.

you have melted internal wiring, probably some mice ate it. call an electrician and DO NOT USE IT!! turn off at the wall/fusebox

You didn't state whether you used a DCS conversion kit or not. I was able to find conversion instructions for a model similar to yours (not an exact match) but it will give you an idea as to what all is involved.
DCS RGSC 305 Instructions Manual

You probably have a bad ignitor in your oven. The gas will not turn on until the ignitor glows red hot - less resistance and more current for valve to open. This can be replace by a repairman or a tool savvy homeowner. If you call a repair man, your looking at 150.00 to 200.00. Part can be from 50.00 to 110.00. I usually charge 110.00 plus the price of the ignitor.

Bad oven sensor or ecu"clock"