I know that a lot of navigation by air and sea in the U.S. references speeds and distances based on imperial measurements: knots (miles per hour), nautical miles, etc. Does the international air travel (and air freight) industry still use that system, or is there a metric equivalent (air kilometers per hour, perhaps?). What about seafaring?
Nitpick: nautical miles and knots aren’t exactly imperial units (unless you’re thinking of the Babylonian Empire, which is responsible for the uncouth division of a full circle into 360 degrees :))
I don’t know about weights and such, but altitudes for airplanes is assigned in thousands of feet. And all international air traffic control is in English.
A nautical mile is not the same as an imperial mile. The only thing they share is the name. You could argue that a nautical mile (nm) is closer to a metric distance in concept than an imperial mile - as a nm is one minute of arc as measured around the equator. Thus it is 360 * 60 = 1/21600 of the diameter of the earth (at the equator)- whilst a kilometer is 1/100000 of a quarter of the diameter of the earth (as measured at a meridian passing through Paris). If the Earth was a perfect sphere these ratios would be exact, but since it isn’t there is a bit of additional error.
Navigation is still often done in degrees minutes and seconds. Locations are still commonly provided as such. However decimal degrees make life easier for computation, and there is a lot of decimal degrees as well now, given just about any computer based system will use them.
However no matter what, if you are navigating on a sphere, your location needs to be in spherical coordinates. Assuming a spherical Earth nautical miles worked out as an easy and comfortable distance measurement when working in minutes and seconds. With an oblate spheroid, you need to do a lot more calculation - which is what computers are for - but nautical miles are still a good distance metric.
There is a standard definition of the shape of the Earth used for navigation and any other positioning - WGS 84. It defines the shape of the earth as a set of spherical harmonics (sort of like a Fourier series but in 3D) and defines the coordinate system. You do need to be careful, as individual regions will define their own datum for local mapping, however GPS receivers will usually work this out is needed.
Nautical navigation is done with distances in nautical miles with speed in knots.
The majority of the world uses knots and nautical miles for speed and distance, and feet for altitude. There are some differences between countries. You use what you are used to until you get to a country that forces you to do it their way.
Mostly navigation can be done using whatever units the flight crew are accustomed to using. Differences only become an issue when communicating with ATC or other aircraft, for example if you flew into Russia your altitude clearances would be in metres and you’d have to convert to feet.
Ugh, I just noticed a stupid mistake in my post above. Wherever I wrote diameter of the Earth, read circumference. 
360 degrees is very far from uncouth. It is a small enough division to be useful and a local scale and divides further (by 60) for global measurements. 360 can be subdivided by 24 different integers (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180) very useful when doing geometry.