When attempting to start the vehicle the engine revs high and then it shuts off

This sounds like it is a theft issue as that is common when it is an active security event. Is the theft light remaining on?

https://www.2carpros.com/articles/how-to-reset-a-security-system

If not then we need to check the fuel pressure as it may be loosing its prime when you start the vehicle. So let's use this guide and test the fuel pressure after the engine starts and stalls:

https://www.2carpros.com/articles/how-to-check-fuel-system-pressure-and-regulator

The last most likely cause is the idle control has failed.

https://www.2carpros.com/articles/how-to-replace-an-idle-speed-control-motor-iac

Below you will find the detail about your idle control. If nothing else, I would replace this and retest the issue.

Let us know if you have questions. Thanks

The lights issue sounds like a different issue so let's get a new post started for that but if you can clarify what you are touching in the relay cover to get the engine to start that would help.

It sounds like there is a loose relay so we need to narrow this down. If you can notice what you are touching to get it to start and then get a picture of that or tell me what it is and we can dig further into that.

Let me know. Thanks

That is great info. So what you are doing is closing the relay yourself rather then the coil side of the relay where it energizes when power is applied to it and it pulls the contact closed.

This means that highlighted wire is not supplying voltage. So either the alarm relay has failed or the ignition lock switch (on manual transmissions) has failed or the BCM is not grounding the alarm relay.

Let's go to the ignition switch and test for voltage coming in. If you have it coming in on the green wire then you can just the switch and the engine should start. This proves the switch is the issue.

Let's start with this and go from there.

Let me know what questions you have.