Is the sun flipped on the other side of the world? As the orientation of the moon is different in the northern and the southern hemisphere, is the orientation of the sun different in both hemisphere? 

Does a sunspot appearing in the Sun northern hemisphere from the Earth northern hemisphere will look as located in the Sun southern hemisphere from the Earth southern hemisphere ? 

If I understand well, an equatorial mount can compensate for the Earth tilt, but how is possible to compensate from one hemisphere to the other one? Is the image flipped?

thank you!
Yes, the orientation of the Sun will be different from the Earth's northern and southern hemisphere, just like your example of the Moon.

I would not say that a sunspot in the "northern" hemisphere would appear to be in the "southern" hemisphere just because of the change in orientation. North is fixed on the Sun and Moon, just like they are on the Earth. (If you turn a globe of the Earth "upside down", the northern and southern hemispheres do not switch, just the orientation."

The view through a telescope may change the orientation, but the directions North, South, East, and West on the Sun and Moon remain the same. The user of the telescope needs to determine which directions those are (up, down, left, right). It changes with the design of the telescope (reflector versus refractor) and the number of optical elements.