Electrical assistance
you have two electrical systems, a 12 volt system and a 120 volt system. Microwave, electric receptacles, air cond, etc (there are others depending on particular appliances in your unit) are 120 volt; they will only work when your trailer is plugged into a 120 volt power source - period.
Lights, water pump (generally), refrigerator (depending), furnace (usually), exhaust fans, etc are 12 volt. They will operate when a properly charged battery is installed, when connected to a vehicle with a 7 pin connector (a 4 pin won’t provide any electrical for the 12V systems), or when the RV is plugged in (as noted above) with a properly operating converter.
My guess is you are not plugged into a proper 120 volt source and therefore the 120 volt system is not operating (the converter is a 120V unit). Any 12V appurtenances that are operating (you mentioned lights) are operating off the battery in the RV (until it goes dead).
Running lights, turn signals, marker lights, etc are connected to the vehicle 12V system via the 4 pin connector. Nothing else is connected to the vehicle 12V system via a 4 pin connector. A 7 pin connector will provide 12V to the RV (when properly installed.
I would recommend doing some research and getting a good understanding of the particulars of the 12V and 120V system and how they intertwine. For example, the fuses are for the 12V side; the 120V will be protected by a number of circuit breakers.
Nickthehunter touched on most of what's possibly wrong. My addition is that most of what you say is not now working is on the 12 volt side of things. You said you removed the old battery but you did not say you replaced it. I would suggest replacing the battery even if temporarily to see if things come to life. But be sure that you know and follow the polarity of the wiring so you don't start blowing fuses and such.
As far as the refrigerator it depends on what you replaced it with. Is it a household 120volv, a 12volt compressor fridge, or a Propane/Electric fridge?
As far as the taillights it's possible you got the wiring reversed. LED's are one way meaning if you hook the wires up backwards they will not work. I would pull the lights and verify that the positive from the trailer is going to the positive on the lights and that the ground is hooked to the ground wire on the taillight.
