Is 18 mm enough for video?
What you're referring to is the aspect ratio. 1920 by 1080p is HD video, you would have to letterbox it in post production to achieve the 'wider' aspect ratio (I'm assuming you want 2.39:1).

As for the lens, 18mm is a pretty wide lens. If you have a 35mm sensor you may experience some barrel distortion – which isn't necessarily a bad thing, just as long as it's used well. 

Not too sure what you mean by your audience feeling claustrophobic. However, a 50mm lens on a 35mm sensor closely replicates the field of view of the human eye as does (approx.) a 30mm lens on a crop sensor. 

So to answer your question: no, you don't need a lens wider than 18mm to make a professional looking video. 

EDIT: You mentioned that you will be using a kit lens. Typically kit lenses have a low build quality and are rarely used by professional cinematographers. Obtaining a better lens will make your videos look more professional due to the superior build quality and glass. 

A good way to look for good lenses is to, firstly, figure out what focal length you require (8mm fisheye, 50mm, 85mm etc.), search online for different brand's lenses (you have  a Nikon but there are many companies that make third party lenses such as Tamron, Tokina, Sigma), look online for reviews of the lenses and test footage (Vimeo is a great place to find test footage) and then you can make an informed decision on which lens is best for you. 

I'd recommend obtaining a good 50mm prime lens. Primes have less glass and generally produce a sharper image. Not too sure what Nikon lenses are like but you could try purchasing a 2nd hand lens off eBay and then buying an adaptor to get it onto your camera. This option is usually pretty cheap and if the lens is in good condition you will most likely be impressed with the result. I bought myself a Carl Zeiss 50mm f/1.8 for $80 and an adaptor for $5 – now it's one of my favourite lenses (and definitely the cheapest).