It should be accurate. That said, I've never used Ai templates. Standard business card size is 2" h x 3.5" w ( horizontal ). That doesn't mean you shouldn't use Ai templates, it's just a matter of personal preference. You shouldn't have to adjust the template. What seems off about the template?
Yes of course the business card template is correct at 3.5"W x 2"h. There are multiple artboards depending on if you choose a folded or vertical, so delete the other artboards.
If your color goes to the edge of your design, you will want to add bleed (standard is 1/8" on each of 4 sides, but some printers may go less to save paper).
Check your printers website for templates, as if you use their template, you will make their job easier and get better service.
Here's a more in-depth look in printing out business cards from illustrator. It's not exactly what you asked but for sure you can find some of the facts and explanations helpful to your printing. http://www.greatfxbusinesscards.com/illustrator.htm