No crank no start, codes P1800, P0340 and P0345. Since P0340 and P0345?

It's a bit tedious to get the starter out. I will do it soon but before getting it out, I am thinking of testing the starter in place by disconnecting the harnesses that connect to the solenoid. Then, I will use clamps from the new battery that's sitting outside the hood to connect directly to the solenoid and then use another clamp to go to the S terminal and see if I get any action from the starter. I just want to make sure that this will isolate the starter and not affect any other electrical parts in the car. Or do I have to take the starter out before testing it? I do not have access to the starter if the battery is in the right place. Thank you.

Thank you for your advice. I connected the battery and measured the voltage on the terminal with the heavier gauge wire. It was 12.5V. Then, I had a helper turn the ignition to start. The car did not start. Then I had the voltmeter connected to the smaller harness. When the helper turned the ignition to Start, there was a momentary 10.1V seen but it then quickly went to zero before the ignition was shut off. I also bridged the large and small harness connections with a screwdriver while the ignition was turned to start. There was a click, but the car did not start. I will be removing the starter, test it off the car, and replacing it with a new one. Thank you.

I thought that the fact that the starter did not work when I jumped the terminals on the starter solenoid and heard a click, but the starter did not crank and roll over is because the starter itself is bad. Is that correct?

I finally succeeded in removing the starter after trying many tools. I then bench tested the old starter. By applying 12V to the main terminal. I then use an alligator clip to apply 12V to the signal terminal. Nothing happened. No sound, no clicks. So, I figured the old starter was indeed bad.
I then took the new starter and performed the same test. In this case, I heard a click and the motor started turning and stayed turning smoothly until I removed the alligator clip.
I then installed the new starter. After that, I used jumper cables to connect the battery standing outside the car to the battery posts in the car. I measured the voltage at the main post of the solenoid, and it was 12.58V. I then had a helper start the car. It made a rapid starting sound as if the battery was bad. I also measured the voltage at the signal terminal when the ignition was turned on. It was only 2V. I don't know what is reducing the voltage at the S terminal. The sound made by the new starter when I tried to start the car is enclosed. I also tested the battery with a battery tested and it tested good. Also, the battery connectors are clean. Kindly advise on how to proceed.

The battery was outside the car, but I used jumper cables to connect the battery to the post connectors on the car. So electrically, it was like the battery was in the car, but physically, there was nowhere for me to place the battery and at the same time expose the starter. By double check the relay, do you mean that I should check the starter relay works?

The problem is now solved. As you suggested, I took out the jumper cables and connected the battery directly to the connectors. After that the car started. Thank you very much for walking me through this long process. Overall, the faults were a bad battery and a bad starter. Once again, thank you.