Hazard lights flash when key in ignition, take key out and start engine again they go off

Insert key into and remove it from ignition more than six times in 10 seconds. When the hazards lights will flash twice insert key into ignition and turn to ACC position.
Then push any button on key fob once, the hazard lights will flash twice. Then open drivers door to END.
Should be all working again.

Since there are two separate problems that occurred during the battery transition, try as we may, we all make mistakes. Go over the wires that you soldered again. I know you already did that but look for two wired similar. But with one difference. More aptly...a green wire with a black tracer, and a black wire with a green tracer. Or a similar scenario. It's a subtle difference. But the old adage of killing two birds with one stone. Probe the wires with the ignition off to see which one has constant power (going to the hazards) and look for it's counterpart.

No, its not supposed to happen. It sounds like you have a fault with the alarm system. When you open the door, it thinks its being entered illegaly, but when you put the key in the ignition contact, it recognises that the key is correct and will allow you to start the engine. You should take it to the dealer to check for the fault. Has the battery been disconnected recently ? or has the battery been replaced in the remote handset ?
It may be that the system requires resetting. To reprogramme the handset, do this -
obtain all keys
switch ignition on and off 6 times within 10 seconds
after 2 seconds, indicators will flash twice
press and hold unlock button
press lock button 3 times
release unlock button
indicators will flash once
if required repeat button procedureto programme remaining key remote handsets
switch ignition on
indicators will flash twice
remove key from ignition switch
check operation of all programmed key remote handsets

That should do it, if not then as i said you would have to go to the dealer.
Hope that helps

Most likely it is the ignition switch (the key ignition switch) in the steering column ($100 at a dealer).

It may be a loose negative battery ground cable, too.
Check the negative battery cable where it conencts to the engine ground. It may be loose, corroded, damaged, or otherwise not making a good ground contact.

Add a grounding strap ($11 - AutoZone) from the engine to the chassis. That may help, but it's most likely the ignition switch.

Sounds like you have an aftermarket alarm with starter disable. By disconnecting the battery the alarm reset and the defualt setting is usually to set the alarm automatically (active arming). You have three options..
A. If you have the key fob that went with the alarm you can disarm the alarm.
B. If no key fob is available you can disconnect the battery again, turn the ignition key to the on position and reconnect the battery. This usually doesn't allow the alarm to set and allows the car to start.
C. Remove the aftermarket alarm system. This is usually the long term fix i recommend if the fob isn't available. Option B should get you going though.

check body control module fuse . check head light relay and the drl lights relay.

you may have had a bad connection at the battery that would cause this kind of problem

Close all vehicle doors, LOCK doors using drivers side switch.

Insert key all the way into the ignition, (do not turn) remove key.

Repeat this step 5 times. (6 total)

Hazards should flash to indicate program mode.

Press any button on transmitter to be programmed, hazards should flash to indicate successfull program.

*For additional remotes, cycle drivers door lock UNLOCK, LOCK, press any button on remote, hazards should flash. Repeat step as necessary.

Ignition ON, (engine off) ignition OFF, UNLOCK using drivers side switch, test remotes.

Source

You need to program the remote. Get in to the car and LOCK all doors with the switch on drivers side door. Take the key and insert it into the ignition cylinder in and out 6-10 times, until the flashers/hazards flash on your dash board. Or just keep putting the key in and out (nice and easy now, not fast) until you see the lights on dash flash. After the lights flash, turn the key into ACC mode to make the whole dash light up.(do not start car) .With the key in ACC position, hit any button on remote, that will make the hazards flash again. Turn key off and unlock door with switch again. Open door.DONE.
Good luck.