Is this watch authentic? If so how would I know?

Everything looks authentic and is marked/branded clearly and correctly, including the movement which is clearly visible through the exhibition case back. Plus those attempting to profit through production and sale of fake watches are probably better served in focusing their efforts on much more expensive and storied watch brands and models than Timex (e.g. Rolex, Omega). I mean, how much profit is anyone likely to make by faking/selling a sub $300 watch that probably isn't in high demand and selling like crazy regardless?

For replacement band/strap, what is the size in mm?

The band lug width is 20 millimeters.

I have a few Timex watches and the lume is pretty much non existent. How's the lume and how long does it last?

It depends on the watch style, some last up to 5 years some are 2 years. If the Indiglo is no longer working, then the battery should be replaced. If the battery replacement did not fix the issue, there is an issue with the watch itself. I would recommend visiting your trusted jeweler and have them check the watches.