Only get air flow from front vents no defrost

The mode actuator is on the top of the HVAC case on the drivers side of the housing up over the center console area. The one you changed is the blend door that controls temperature. The mode door uses an actuator and a plastic cam to move the various doors and flaps to direct the air and that plastic has a bad habit of breaking.
To get to it you need to remove the floor duct on the drivers side, then the console if equipped and the console duct. Then disconnect and remove the actuator and cam as a unit.
To install the replacement you reverse the process but need to be careful to
- Line up the heater/defroster valve lever with the mode actuator cam slot.
- Line up the A/C valve lever with the mode actuator cam.
Then test it before you put the other parts back on.
The part number for the replacement with the cam is - #52473356 and looks like the last image. You can get just the actuator but most times it's the plastic that breaks.

I could but you could probably buy another truck for that house call charge. LOL

The hardest part is usually getting comfortable under the dash! For that I made a simple creeper that slides in on the truck floor and onto an adjustable sawhorse outside. Makes it much easier to just lay back and work. Oh and one of the head strap lights is a big help.
Good luck with it.

With a console the book time is 1.4 hours. Without it's under an hour. That is because the console can be a pain at times. The reason to remove it is to get access to remove the heater duct, I have done it without removing the console but you end up needing to fold your arm into interesting positions to get into the area needed. I would look it over first and see what you think. One good thing is that even with it out or partially apart it wont prevent you from driving it if needed or from just deciding "I've had enough, it's time for a break".

That relay sends power to the rear glass and mirrors if it has that defrost option. Nothing in the HVAC gets power from it. The attached is the wiring diagram for your HVAC system showing the actuators. The defog relay is turned on just for the high current draw of the wires in the grids. If your truck doesn't have that option then it's a spare relay.
However it sounds like you have other issues now.
If you need help with the no-start just ask as a different question please, we like to keep different issues as individual items to help when searching later.

You can but I doubt it will do anything. That relay is only for the optional heated glass. A quick test for the HVAC does the blower work on all speeds? Can you select the temperature of the air. These are both with the key on. If those don't work either then check the ignition A, B and C fuses in the fuse block under the hood. They provide power to the interior parts through the ignition switch. Then to test if power is getting through the ignition switch take a simple test light and connect it to ground so you can reach the fuse panel on the end of the dash. First turn the key on, do the turn signals work? Next do the wipers work? Next use the test light to see if there is power at the cruise fuse, if there is then check if there is power at the Ignition 3 fuse in that same box. Next check for power on the ignition 0 fuse and finally for power at the 10 amp crank fuse with the key in the start position. If those all test good the ignition switch is okay and isn't the part linking the two problems.
Then if you ask about the no start at - https://www.2carpros.com/questions/new
I can continue that issue as well. That way the site owners are happy. Plus I can post the wiring diagrams for the starting circuit.
Now if any or more than one of the power tests fails note which ones and post that, for the HVAC issue the only ones that matter are the ignition B and Cruise fuse powers and if the blower works and you can change the temperature of the air, that would be the actuator you replaced the first time.
All of the above will also help with the no start and the heat issue. And as a bonus you only need a test light and not to take anything apart, yet.
Test light wise any of the cheap 12 volt bulb types will work, nothing fancy needed.

Okay, hit that link and give the same info about the truck, that way I can give you some areas to check to see if it's a relay, fuse or worse. You can toss my name in there if you like but there are a few folks on here.

That isn't that bad. Actually with the batteries they make these days it's pretty common. The internal plates fail and you get voltage but no usable amperage. It's one of the gotchas when it comes to using a meter and why I have about 5 different test lights all that can load a circuit for testing. The meter would show full battery voltage as long as one strand of wire is connected, but if you put a load on it you discover there is no ability to carry that load. It's why one of the first things I would have asked would have been to pull the battery and get it tested, many of the parts chains will do that free.

It's getting to be a difficult thing to pick a good battery these days. There are only about 3 companies still making batteries and those keep trading hands and plants faster than I can keep track.

There are 3 actuators total, one behind the glove box and two on the opposite end of the case. If you changed out the upper one on the drivers side that has the large cam attached to it but the defrost still doesn't change over when you select it and you can see the mode actuator moving that cam and the door shaft, then it's possible the internal door itself has failed. For that the entire dash and HVAC case needs to be removed and the door repaired. I don't recall seeing one fail but it could happen, normally the issue is with the cam and actuator though.