First. The neck looks bad in both cases to me.
very turtle like.
I asusme your model is not like that?
second
Eyes are basically always separate. So is the jaw and tongue.
however when you import the model they can be just 1 mesh.
You would have to make them children (parent them) to the body.
And no. For performance sake they should definitely all be part of the same mesh.
I have never used auto rig, but in general the export settings are the most important thing to worry about.
Also, on import you need to either select the correct skeleton or add a new one.
adding a new one to the messed up first model may be enough to bring the parts back to their proper place.
As far as IF you should attach in einge or in 3d program.
it depends on if you plan to have the character remove it or not in game.
Look into Modular characters for that. Usually clothing is baked on as well.