#1: You should make sure that you are using the same, up-to-date Inquisit 4 version on both the Mac and the PC. The latest release is 4.0.9.0 at the time of writing, and it's available via
https://www.millisecond.com/download/ .
#2: > The timing of stimulus presentations is different
Please be more specific here if possible: How is the timing different and what is the respective code (in the <trial>'s /stimulustimes or -frames attribute). Especially if you are using /stimulusframes, different timings are to be expected if the two machines operate at significantly different display refresh rates.
#3: > as is the size (on the screen, that is) of the .mov files and images.
If the two screens in questions have vastly different resolutions, physical dimensions and aspect ratios, yes, the layout will necessarily be different. Usually, you will want to specify all sizes and positions in terms of percentages. In addition, you can make use of the <defaults> element's /canvassize and /canvasaspectratio attributes to achieve greater consistency across screens with different sizes and particularly aspect ratios (e.g. a 16:9 wide-screen vs. 4:3 display). Virtually all of the scripts available in the library make use of this and can serve as examples if needed.
#4: > The .mov files also tend to show up on the PC along with whatever is on the screen in the background [...]
MOV is not an ideal format for use on Windows PCs. You may want to consider converting the videos to either a more Windows-friendly format (e.g. WMV) or to a format that is relatively well-supported on both platforms (e.g. MPEG2). Those format considerations are covereed in the language reference for the <video> element in the Inquisit documentation.
Hope this helps.