Olympus e-PM2 - what am i doing wrong?

The reason why your settings don't make sense to me is that the camera has picked ISO 800 for bright sunny scene (which it would not do normally), picked ISO 200 for dark scene (which it would not do normally) and some of your photos say 'manual exposure' - which means YOU have set the exposure, some have auto white balance and some have manual white balance.  None of that implies the camera is working as it should --- which is probably from your unintentional user error somewhere.

The reason why people sound frustrated with your answers is because experienced photographers know the camera is not responsible for taking pictures, the operator is.  The more advanced the camera (and the EPM2 is actually an advanced camera), the more responsible the operator has to be.  To imply the camera is either 'really good' or 'really bad' is like saying the pots and pans were 'really good' to make the dinner, or the paintbrushes were 'really good' to make the painting.

So you do have to decide whether you want to learn something about photography or whether you do not.  It isn't actually difficult, you just have to accept that you must learn something.  And you have to spend time getting the camera set up properly --- this camera is actually a bit confusing to set up.  There are tons and tons of settings that can be configured (remember this is an advanced camera) and some of these settings are initially hidden by default.  So you may need to spend a LOT of time figuring out where these settings are --- use the manual.

Then start with these easy steps:

- Set the camera on P mode (Program mode, it will pick your exposure settings for you). Make sure you have ISO set to auto and you have set the upper limit for ISO to at least 6400 or higher.

- Make sure stabilization is on, setting #1, remember to hold still when shooting

- Use face detection if you are shooting your kids, otherwise turn it off

- Make sure focus is set to S-AF, and make sure you are using ONE focus box, set where you want to focus

- Make sure white balance is set to auto (if you do not want an orange color indoors from the lighting, there is a menu setting to turn the 'warm' color off).

- Use picture mode 'natural', default settings for sharpness, contrast, saturation, etc. Use gradation 'normal' for now. Later you can experiment with auto gradation for shooting when there are lots of shadows.

Now go out and practice.  Start with outdoors pictures until you are comfortable with what you are getting.  Then shoot indoors but have your kids pose for you.  As you get more familiar with things, start some reading to learn how to better take photos indoors.  It isn't hard and it isn't fiddly and you do NOT need expensive primes to get good photos indoors.  Even of running kids.  You just need to know how to do it and the best way to learn is to practice and figure out why each and every photo did NOT work.

And if this seems too much work for you, return the camera and use your phone!