Yes, a 3/8" buckle will work with this project.
The ITW Nexus buckles (various sizes/models) are my favorite, but they're quite a bit more expensive than the many cheaper imported versions, which work well enough for most folks, so it comes down to what you want to spend. Some folks only want actual mil-spec paracord instead of the cheaper quality commercial paracord, and they don't mind spending twice as much for it, again it just comes down to your choice. ;)
It can take time to untie/unravel a knotted paracord bracelet, and the cord will probably be kinked up, and may be dirty/worn and weakened depending on how hold it is and how often it has been carried, but can still be useful if needed. There are many different possible designs for paracord bracelets, some use patterns than can come undone quickly, like a chain sinnet. Considering that many paracord bracelet designs use around 10 feet of cord, some more, some less, if someone has an urgent speedy emergency need for that small amount of cordage, then their problem probably is beyond a basic utility need that length of cord might assist with... ;)
Width will vary by the actual cord you use, and paracord can vary in thickness/width of the strand from one manufacturer to another, and how tight or loose you tie the bracelet/watchband. The tan one in my photo has a tied width of about 7/8" thick.
Maybe. It would depend on how flexible the wire is.
Different gauges of wire are often used for jewelry making (braided/twisted/knotted/knitted). The thinner wire is usually more easily bent and worked where the thicker it gets may require tools to bend and twist it into shape.
For watches with thinner attachment sections, I've used gutted paracord, and done two piece watchbands to work with those type watches. You just have to experiment with different paracord designs and options base on the variety of watches out there.