EC2 Instance failure (about system logs and state hanging)

Hello.

When trying to shut down the instance (since SSH can't reach it), it seems to just hang. Force stopping also doesn't seem to actually force the stop. I've tried looking at the docs, but the wording seems unclear to me. Does If, after 10 minutes, the instance has not stopped, post a request for help mean that force-stop is expected to take 10 minutes?

Normally, it takes less than 10 minutes to force a stop.
If it takes more than 10 minutes, there is a possibility that an abnormality has occurred and you need to contact AWS Support or request help from AWS re:Post.

Logs. After recovering the instance, the on-disk logs doesn't show any signs of trouble. The journald entry for that boot just stops suddenly, indicating a very hard crash. Our regular monitoring does not show any signs of CPU (5% usage) or memory (70% usage) exhaustion. Of course, if it's a kernel panic, the logs on disk will not show anything, so we tried looking at the system logs/console output after the instance was recovered, but the text before boot was already scrolled out. The command was aws ec2 get-console-output --instance-id i-0c48fd2b6b069df76. I looked at the docs and help function, but couldn't find a way to scroll backwards. Is there a way to do this?

Even if you follow the steps in the document below from the AWS Management Console, are you still unable to see old logs?
https://docs.aws.amazon.com/AWSEC2/latest/UserGuide/troubleshoot-unreachable-instance.html#instance-console-console-output

Additionally is there a way to log these (the console output/system log) to a cloudwatch log-stream so they become permanent? Or some other way of achieving the same goal?

Depending on your OS, you can install CloudWatch Agent and output logs to CloudWatch Logs.
You can install CloudWatch Agent by following the steps in the document below.
https://docs.aws.amazon.com/AmazonCloudWatch/latest/monitoring/install-CloudWatch-Agent-on-EC2-Instance-fleet.html

Also, as an example of log output settings, the following settings will output "/var/log/messages" to CloudWatch Logs.
Please note that the system log file path varies depending on the OS.
If you are using Amaozn Linux 2023, by installing rsyslog, system logs will be output to "/var/log/messages".

{
  "logs": {
    "logs_collected": {
      "files": {
        "collect_list": [
          {
            "file_path": "/var/log/messages",
            "log_group_name": "syslog",
            "log_stream_name": "{instance_id}",
            "retention_in_days": 30
          }
        ]
      }
    }
  }
}