Yep. that occasionally happens even with a brand new out of the box unit.
Before you do anything else, hook the unit power cord directly to a fully charged, good condition, battery to see if that is the issue.
Screwy power will cause lots of symptoms.
The speed sensor you have no connection for is the speed wheel accessory. GPS speed is what you get when the GPS is working--two different things.
You can look up the flat rate repair charge on the Humminbird web site. Yes Humminbird will fix it at the same place it is made--Eufaula, Alabama.