Check gauges light

Your description of the symptoms is confusing, but I can start out with some observations.

When do you see the "check gauges" light? Is that after the engine stalls? If so, that is to be expected because the alternator has stopped working, so the "volts" gauge will be low, and there is no oil pressure, so the "oil" gauge will be low. The "volts" gauge can also be low if the engine is idling too slowly. In either case, if the "check gauges" light turns off when you speed up the engine, it is likely there is no problem.

Since you disconnected the battery, engine idle speed is going to be too low. That can make the engine not start unless you hold the accelerator pedal down 1/4". You will not get the nice idle flare-up to 1500 rpm at start-up, and it will tend to stall at stop signs. By disconnecting the battery, the learned data stored in the engine computer was lost. Most of that will be relearned as soon as you start driving, except for "minimum throttle". That will be relearned when you meet certain conditions. Drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals. Once you do that, the computer will know when it must be in control of idle speed.

If the battery continues to run drained, and/or the check gauges" light is still on, measure the battery voltage with the engine running. You must find between 13.75 and 14.75 volts. If that is low, have the charging system tested for "full-load output current", "charging voltage", and "ripple voltage". Tell me what those values are in your reply.

I cannot see the "check gauges" light in your first photo, but if the engine is not running, that light is going to be on because the oil pressure is at 0 psi.

The do the preliminary test on your charging system, measure the battery voltage with the engine running. You should find between 13.75 and 14.75 volts.