The ignition key sometimes refuses to turn on, and it seems to be caused by the transmission not recognizing it is in park. Playing with the transmission shifter often results in a click which then allows the ignition key to turn. I am skilled in electrical tracing, so if I know which wiring harness leads the transmission microswitch connects to I can work a solution.

Check the basics first. When you first turn the key do you get the dash lights and do they change when you try to start the car? Ignition barrels do sometimes fail and can show as the dash lights not working correctly.
Open the hood and get a helper to turn the key on and off 2 or 3 times. Listen to the fuse box. Your listening for a quiet clicking noise. Sometimes the Starter Relay can fail causing a no start. These are cheap enough to buy so it might be worth replacing it anyway. I've included a link to a web page for the fuse box diagram.

https://fuse-box.info/chevrolet/chevrolet-equinox-2005-2009-fuses-and-relays

try turning your steering wheel from left to right while you gently try removing the key. if that doesnt work and your car is a automatic with the shifter in the center console there is a cable running to the transmission that sometimes becomes loose causing the car not to be fully into the park position.

key wont come out or turn to off until gear selector is in park [ recognize that its in park ] your selector or transmission selector cable is damaged

Make sure transmission is in Park. Key will not come out until ignition is fully off and ignition will not turn fully off until gear selector is in park

You have to watch that now ,even on a 2006 vehicle . With the electronics an the number of control modules , every system has an electronic control module - a computer . When they lose power strange thing's happen .Sometime the only thing that will get the vehicle up an running again is a trip to the dealer . Battery Disconnect or Replacement Problems There are only a couple of thing's you could check an only if you know how the stating system works .It depends on what model LH2 or LY7 . You have to look in the under hood fuse / relay box an see if there is a STARTER relay . The starter relay is energized by the ECU - engine computer, there by cranking the engine over . Circuit Description
Moving the ignition switch to the CRANK position sends a 12 V signal to the ECM Ignition Crank Sense. The ECM verifies that the transmission is in the PARK or NEUTRAL position. The ECM then grounds the control circuit of the STARTER relay. When the STARTER relay is energized it allows voltage to the starter solenoid S terminal.
The problem of not getting your key out of the ignition probably has to do with the
Electronic Park Lock (EPL)/Ignition Lock Cylinder Control Actuator: If the vehicle is equipped with automatic transmission and a floor mounted console gear shift, it has an ignition lock cylinder control actuator system in the steering column. The ignition lock cylinder control actuator purpose is to prevent the ignition key from being turned to the OFF position when the transmission is in any position other than PARK and the vehicle may still be moving. The column ignition lock system consists of an ignition lock cylinder control actuator, and a park position switch that is located in the automatic transmission shift lock control switch. The ignition lock cylinder control actuator contains a pin that is spring loaded to mechanically prevent the ignition key cylinder from being turned to the lock position when the vehicle transmission is not in the PARK position. If vehicle power is lost, and/or the transmission is not in the PARK position the operator will not be able to turn the ignition key to the lock position and will not be able to remove the ignition key from the column.

Linear Shift Assembly: If the vehicle is equipped with a column mounted gear shift, it has a linear shift assembly on the steering column. The linear shift assembly has a cable that runs from the linear shift assembly to the ignition lock cylinder case. The purpose of this cable is to prevent the ignition key from being turned to the OFF position when the transmission is in gear and the vehicle may still be moving. The linear shift assembly cable contains a pin that is spring loaded to mechanically prevent the ignition key cylinder from being turned to the lock position when the vehicle transmission is not in the PARK position. If vehicle power is lost, and/or the transmission is not in the PARK position the operator will not be able to turn the ignition key to the lock position and will not be able to remove the ignition key from the column.
Given the amount of electronics on your vehicle ,without hooking up a factory grade scan tool to see which modules are still communicating it would be impossible for me or anyone else to give you a diagnostic direction. I have had vehicles like this come to the shop after the owner tried thing's, an it cost them a lot more then it would have if they just had it towed to the shop .You have no idea about the complexity of you vehicle .
The

Replace ignition switch (the tumbler portion) seems like its hanging up, and you shouldn't be able to remove the keys unless the switch is off.The park safety switch possibly. This switch tells the vehicle its in Park. Hope this helps.

it not making up the switch on the shifter on the transmission tell its in park .

Shifting to NEUTRAL
To shift the transfer case to NEUTRAL, first make sure
the vehicle is parked so that it will not roll:
1. Set the parking brake.
2. Start the vehicle.
3. Connect the vehicle to the towing vehicle.
4. Put the transmission in NEUTRAL (N) or have
the clutch pedal pressed for vehicles with
manual transmission.
5. Shift the transfer case to 2HI.
6. Simultaneously press and hold the 2HI and
4LO buttons for 10 seconds. The red NEUTRAL
light will come on when the transfer case shift to
NEUTRAL is complete.
7. Shift the transmission to REVERSE (R) for one
second, then shift the transmission to DRIVE (D)
for one second or FIRST (1) for vehicles with
manual transmission.
8. Turn the ignition to OFF.
9. Place the transmission shift lever in PARK (P) or
FIRST (1) for vehicles with manual transmission.
10. Release the parking brake prior to towing.

************************

Shifting Out of NEUTRAL
To shift out of NEUTRAL:
1. Set the parking brake and apply the regular
brake pedal.
2. Start the vehicle with the transmission in PARK (P)
or FIRST (1) for vehicles with manual transmission.
3. Press the button for the desired transfer case position
(2HI, 4HI, AUTO 4WD or 4LO).
4. Put the transmission in NEUTRAL (N) or press the
clutch pedal for vehicles with manual transmission.
5. Shift the transmission lever to the desired position.
After the transfer case has shifted out of NEUTRAL,
the red light will go out.
6. A re-engagement sound is normal when shifting out
of NEUTRAL.

if you have isolated the problem as i think you have: remove the shift lever cover and adjust the neutral start switch

You will need to most likely replace the lock cylinder and the ignition switch, sounds like a worn out parts issue. This should cure your problems with the lights and any other strange electrical problems.