Yes cops can see a suspended drivers license in their computer system.
If the cop goes back to their vehicle and runs your license they can tell if the license has been suspended, revoked, cancelled etc.
If the license is revoked or suspended, canceled etc then you will be arrested and taken to jail because driving with a suspended license, revoked license etc is an arrestable offense.
So if you get pulled over driving on a suspended license then expect to get arrested and go to jail.
If you get pulled over with just a learners permit and nobody else is in the vehicle with you such as a licensed driver 21 years or older then you could get ticketed.
In most places the law requires that a driver with a learners permit have a licensed driver 21 years of age or older riding along with them.
If you do get pulled over without that driver and you're under 21 yourself then you might get your vehicle towed or get a warning depending on the laws in your state and officers mood.
But if you just got pulled over for something minor and you have a licensed driver with you then the same thing will happen when you have a regular drivers license.
You just show the officer your learners permit, insurance and registration and then the officer will take it from there.
As long as you were not doing something bad then you will be let go.