You have a wiring issue causing your parasitic battery drain, light issues, and A/C and anti-theft issues. You'll have to get creative with a multimeter to follow the lines to see where the draw/short is at.
Replace the DPFE sensor. They go for $50-80 at your local auto parts store or the Ford dealer. The DPFE sensor controls the EGR valve. The EGR valve should open under light load at higher speeds to decrease emissions. You can check the operation of the EGR valve, too, by putting a constant-vacuum hose to the port on the EGR valve to open it at idle. When vacuum is applied, it should shudder momentarily and then get back to nearly normal RPMs as the ECU compensates for it.
The shudder can be from bad gas, dirty fuel lines and injectors, clogged fuel filter, worn plug wires, dirty/worn spark plugs, and the transmission.
I'd first try to determine if it's a misfire from the engine or the transmission torque converter shudder. TC shudder usually occurs when the ATF is out of spec (old and dirty). Check your ATF level and quality. If it is low top it off. Remember to check when the ATF is HOT, after you had cycled through all of the gears, and with the engine running in Park. If the fluid is brown or smells burnt, change the fluid. Remember to drain the torque converter and not just the pan to get as much fluid out as you can!
While you're down there you should be able to adjust the TV cable which links the shift lever to the transmission to tighten it up.