If the oil light is working and is off with engine running, then you have oil pressure. To determine how much pressure, you need a pressure Gage.
Check
Check the brake light bulbs, Check the brake light fuse, Check the brake light switch, In that order.
If the light is not coming on when you turn on the ignition, check the bulb, it may be blown.
No
Oil level or pressure will not trip a check engine light.
The light will reset itself if the system is working properly and the tires are at the correct pressure. If the pressure is correct and the light is still on, you need to take it in for service.
Never seen a check engine oil light. There is a low oil pressure light which means you have low oil pressure which could be low oil level among other things.
There is no such thing as an oil pressure light on most cars. Some cars have oil pressure gauges, and all have a "Engine" light or "Check Engine" light with a picutre of either an oil can, or Engine Block The oil pressure warning light usually looks like a little red oil can. The check engine light looks like a little yellow engine.
It means at least one of your tires pressure is too deflated. Check the pressure and inflat the tires to the correct pressure and rest the light.
Not necessarily.
If you dash lights are not working check that your parking/brake lights are working also check fuses for cig lighter and interior lights that may help get the dash light working again.
There is no reset for a check gauges light. If the light is on, there is a problem either with the temperature oil pressure or voltage.