2007 rav4 instrument panel light on after key out

I replaced battery on 2007 Rav4 two weeks ago, and yesterday when I turn car off the check engine and Low oil light staid on. Car did not start battery dead today. I started car with a batery starter and had a frined put a computer decoder to see what problem is. Computer does not diagnost any problems. Know instrument pannel goes crazy when I take key off. Ther is also a noice in engine when car if off. Like a sensor or computer is using energy somewher in engine. Help will be greatly appreciated. I want to take car to dealer, but want information so that they can not BS me.

Does The Vehicle Have Remote Start ? Why Did You Replace The Battery ?

There are a number of electrical problems on 07 Rav4s, but one that may apply to this one involves deficient Remote Start ECUs creating a parasitic draw.

Who replaced the battery ? Did they know what they were doing or bumbling through the job ?