BLUF (Bottom line up front) - I'm going to assume you have a wiring problem vice everything below so that's where I would concentrate my efforts by tracing and performing continuity checks on those lines. The remainder below is just in case...
Like OldTexasDog recommend starting with the basics. Ensure thermostat and water pump are working. If you have a tuner (like an SCT) pull codes to see what they say. If not, see if you can borrow a reader to get the codes.
Sounds like the BEC out of the V6 may be introducing new problems. You might want to fix the burned connector in the old BEC and start the troubleshooting process again.
Reinspect the BEC to ensure no other issues with the connectors and chase/perform continuity checks on any suspect lines (burned/corrosion/etc); each of the low/hi speed relays throughout the fan circuit. From your description it sounds like your fan was failing and/or the grounds were creating high resistance (hence, melting the relay connectors). Check/clean Ground connections and perform continuity checks from ground points (see wiring diagrams) to battery negative especially as there was obviously a high current condition previously. If those continuity checks are not less than 1 ohm (should actually be like a tenth of an ohm) I would chase and clean all of the ground points in the circuit and redo them until they are even that means adding ground lines. If the problem however was high resistance in the old fan motor that may be resolved.
Now that you've replaced the fan you need to test what part/s of the circuit is working or not. If you have wiring diagrams manual see pg 33-1. If not, they're online:
http://iihs.net/
There's are several ways to test the fans/fan-circuit. A good explanation of the circuit is here:
1) With the motor running activate the A/C and check if the fan is on or not. If it runs that tells you the fan works at least. If that doesn't turn the fan on (turn the motor off) disconnect the main connector to the fan and run a couple of jumpers for Ground and Low/High to the battery to test the fan. Looking into the fan connector (where it connects to the plastic fan shroud) there are 3 pins. A single one on the lower left and 2 on the right. The single one by itself is ground. The two on the right are for low/high. If you jumper it and it runs you know the fan is good at least.
2) If someone/dealer has the proper test equip/monitor they can command the low/hi speed outputs from the PCM to come on. If they do the circuits work.
3) If you don't have that piece of test equipment (usually a dealer only item) another way to test is to get yourself a 4-post relay tester (
) and plug it into the fuse box and plug the relay on top. Apply a ground to the low speed relay ground input (lower right pin); repeat for the hi speed relay; if the fan runs the relays and the majority of the wiring circuits are good. If you're unsure if a relay is working you can substitute the horn relay (as long as the horn works). Now, you need to determine whether the PCM is providing a ground out on pins (6,7 of C175; PCM) to (C11, A12 of C1035B; BEC) (See wiring diagrams manual see pg 33-1) when it reaches temp settings to run the low/hi speed relays.
4) If you have software (like Logworks or similar) you can monitor the PCM signals and determine when/if they activate.
5) If you have someone who writes tunes for you may want to ask them to create a fan-test tune that will turn on both low/hi fans at a ridiculously low temp (say 40 degs). If the fan runs (with the car running) the circuits are good. Test each relay for operation by pulling the other. Then reload the last good tune.
I've asked Lito (Manuel) where the temp information comes from for the PCM output. His response below:
Q1: How does the PCM get it's temperature information? Where does it come from?
Q2: Does the Cylinder Head temp sensor have anything to do with turning the fans on/off?
Lito:
OK, the ECT temp is inferred as there is no actual sensor for that (in a 3V).
The ECT temp (even inferred) is used for FAN activation points.
The only temp sensor is the CHT that is screwed into the passenger's side head.
Folks, if I'm missing anything major someone feel free to speak up.