Using a 3-meter DEM, can I create an accurate 5 feet interval contour file? Or is the minimum interval contour I can get is 10 feet (roughly 3 meters)?
Picking the contour interval has less to do with horizontal resolution of the DEM, 3m in your case, and is instead mediated by the accuracy of the elevation values.
Specifically, to produce valid contours without forcing the interpolation software to "guess" more than it reasonably should, the smallest interval you can make contours of is limited at double the vertical error of the dataset.
You'll have to check the metadata of your DEM, but I expect somewhere you'll see it report something like "vertical RMSE = +/- 30cm." In this case, one would be limited to making contours of 60cm or ~2'
The answer depends on your landscape, but generally your contour interval can be much less than the resolution of the DEM. Are you looking at a normal flat-ish or hill-y area, or is this something like a Rocky Mountain range with very, very intense vertical change?
If you think about vertical changes versus horizontal distance, there is generally a much lower rate of vertical change in values per horizontal change, for most landscapes.
Using a 3-meter (lets just say 10 feet) resolution elevation raster in a lot of cases means that the elevation change from pixel to pixel is going to be far, far less than the size of the pixel. In other words, for every 10 feet, you might only go up or down a fraction to a few inches on average across a landscape. At a smaller scale -- maybe you're looking at a particular hill, or a urban landscape -- you may have some intense slopes or straight drops in places where you might require a higher resolution DEM to capture those contours more accurately (if that's your purpose).
I think with a 3-meter DEM, you will most likely have good enough accuracy to represent the landscape with contours with an interval as low as half a foot. So 5 feet may possibly too coarse! You would likely be losing a lot of accuracy available in your data. I would recommend 1 foot for general purposes, unless it's a rather large scale area and you don't need that level of detail.