If there is a satellite or the ISS passing by over a point during a total solar eclipse in the darkest region, will the satellite be visible by naked eye or a pair of binoculars, since it will be dark on Earth and the slight sunlight from the uncovered part of sun might light the satellites or ISS up.
As an answer to the question in your title, NASA photographer Joel Kowsky captured this stunning composite earlier today in Banner, Wyoming. The positioning required to perfectly frame Station during this solar transit took months of planning, and the window of opportunity was brief enough that it was shot at 1,500 frames per second with a high-speed camera.
Goddard Space Flight Center has a flickr album that includes video of the transit, as well as a few other details of its capture.
If the ISS was overhead during the Total Solar Eclipse, you could have seen it. During Totality, it isn't nearly as dark as during actual night, only a few bright planets were visible. But the ISS is bright enough that it could be seen, if it happened to be overhead at the right time. You might be able to see a few other really bright satellites as well, but not many.