Your issues seem to be rather in line with what one would expect from a 6-year old getting in touch with observational astronomy. Actually, adults can sometimes experience these difficulties too when they first get in touch with it. I'm speaking out of experience of having shown objects of the night sky through a telescope to visitors. Some people just don't understand that when they touch the telescope, it moves. I would recommend the following guidelines to "fix" your proposed problems (expanding on hartacus' answer, which appeared while I was writing this):
- Someone else could keep her busy while you set up the telescope. Since it is computerized, you won't waste time looking for objects and the only downtime will be due to the moving of the telescope. You can minimise this by planning your observation ahead of time to minimise the distance between two following objects.
- As for the eyepiece, just tell her how to use it. She'll learn it in time. Even adults who have never used a telescope before generally need to be told how to look through it.
- Most telescopes have some mechanism to lock the primary axes of the telescope mount. You can use this to make sure the telescope always points to the same spot. If you find that it is the tripod that is moving, you could try to find a way to lock that as well. Maybe surround its legs with loose bricks?
- To minimise the last two difficulties at the same time, the best position to look through a telescope is to have both hands behind your back, move one eye as close to the eyepiece as possible without touching it, and close your other eye.
As a final remark on looking through the telescope, she might not be seeing the same thing as you when she looks through the telescope, because her eyes are different from you. I don't know whether this is applicable, but most telescopes have a way to adjust their focus to make the image as sharp as possible. People with near - or farsightedness generally require the focus of a telescope to be different.
These, however, are difficulties you will definitely overcome in time. When these are fixed, the most important part of the experience will be in the objects you show her. As hartacus mentioned, show her things she can relate to (planets are great), and tell her about the things she is looking at.
Apart from planets, star clusters can be impressive too. She'll surely be astonised if you tell her how many stars there are in one cluster. Some open clusters can even be thought of as 'micro-constellations' (more specifically, this one).
There are some nebulae / galaxies that are nice to watch, but I would restrain from them in the beginning because most will appear as blurry grey spots through an average telescope, which is not that impressive.
One last thing: you mentioned a binoculars. You can already see a lot with binoculars, although most impressive objects would be too faint. The main issue with the binoculars is that you can't aim them for her (unless you have a mount for them, which exists for some). If you can teach her to aim them herself however, there are some spots in the night sky which have 'mini-constellations': sets of stars in which you can recognize images, but they are too small to see with the naked eye, and too big to see with a telescope.
