I did the same research and I ended up getting an opened trailer.
I started with just the Elise and now, about $12000 later, I have the towing vehicle and the trailer.
Just to be sure, I went to the trailer vendor and I looked mostly at enclosed ones. They only had 20" ones (8.5" wide) which the dude recommended since it would allow more space inside to fasten the car and all that. While they were nice and not too expensive ($5000), I felt overwhelmed just looking at them. My initial reaction was that it would be daunting to pull one of those.
Obviously, there are pros to enclosed trailers, such as safety, weather protection, shade for the car, so on so forth. But in the end, I just didn't see myself tow something that big and be comfortable doing it. But it's just me and I know that tons of people are fine with this.
I talked to the guy and I bought a 16ft open trailer. It's steel and not aluminum, so it's porky. The floor is made of wood, which is exposed to the weather and bird crap (which happens to deteriorate stuff quickly). It's supposed to weight 1600lbs, and it does feel like it.
Fortunately, my truck is equipped with a Class IV hitch, which means that the tongue weight can go up to 1200lbs. I bought a tongue weight scale and it showed that, with the car on top, I was at a tad over 750lbs and moving the car back almost two feet decreased that weight by less than 200lbs (I was at over 950lbs initially).
I got some D-rings which I use to fasten the car and they were quite easy to install since the floor is made of wood.
I went to Sear Point last Monday and it turns out that I did about 14MPG, which is not bad considering that the first few miles are going up a decent sized hill. I'm going to put some truck bed liner on the wood so that it won't be exposed to elements and last a lot longer.
I even have picture
(I didn't buy the truck in that trim on purpose, I swear).