Here is a clumsy work-around that worked for me.
I had the same problem today in Word 2010 on a recently installed Windows 10 computer. When I inserted a picture into a paragraph the Position Object and Wrap Text options were greyed out and I could not affect any changes to the image position/wrapping.
After several attempts to re-insert, cut/paste, close and reopen the document, I finally opened the document on another computer running Word 2007 on Windows 7 and was able to insert the picture and edit positioning. Saved then opened again in Word 2010 and
am able to move the image and insert new images with no problem.
The file was originally emailed to me as a .docx file. I don't know what version of Word created it.
Has anyone found a way to fix this?