Car is beeping constantly while driving

I have had my Mazda Protege5 into the dealer once already because of one very annoying problem, that has now come back. When I start my car in the morning and it is cold (single digits) the car will start and the rpm’s stay at zero along with the mph gauge and the car will beep (like the door is open) but I assure you it is not. Then I put the car in reverse and it drives like normal, but still beeping and gauges at zero, this continues after I put the car in drive and start driving. It continues like this until I shut my car off. The beeping probably lasts five minutes and then stops and comes back in about a minute. This does not happen everytime I start my car, but it has happened about 5 times in the last month.

Also, I forgot to add the first time the dealer replaced the key reminder.

Sounds like a bad connector on the wiring harness from the engine compartment to the dashboard.
Has this car seen a lot of water and salt?

Nope the car is from Arizona, things here tend to dry out and crack.