Svenn, the bad news is that the horror stories (at least those on this website) are pretty much true and correct. Many folks here can respond to specific questions -- but believe us, you'll have some interesting times with a rental car at the salt. It's not impossible -- but you'll need to be careful in a couple of different ways.
First question: No, I don't know if there's a preferred rental company. The one that used to be great - "Rent-a-Wreck" - is no more. Others will state that you're not allowed to go off road, not allowed to take the car on the salt, etc, etc. Some of the issues can be worked around -- like stating that you're renting the car to use on vacation to southern Utah or western Colorado - anyplace BUT the salt flats. Act like a regular tourist (?) and don't be wearing race shirts and hat and so on when you're at the rental desk. Another ploy might include renting the car at the airport instead of at an office in the city, since at the airport they might (note MIGHT) be a little less picky about the questions they need to ask. You might be able to slide under their radar, so to speak.
Second: The closest rental parking area would be Land's End -- that's where the pavement ends. From there to the pits is often 5 miles, to the start line about 3 miles. The pits themselves are about a mile and a quarter long by a few hundred yards/meters wide. Bicycling from Land's End is possible -- but a good long ride, and likely you'll get tired of it in a day or two.
You're likely to be able to leave your bike with Lynda at Land's End. Check with her (she's a member on this Forum) before taking it for granted. But -- if you're satisfied with the informality of another idea, think about arranging with most any race team to leave your bike and other kit at someone's pit. Security on the salt (after hours) is very good -- you shouldn't need to worry about locking things, although it's never a bad idea. Leave your stuff with a race team and hitchhike from Land's End to the pits and back. Once you're in the pits you can bike around and even ride the two miles down to the start line - or you can often hitch a ride with a crew or spectator. I'm not experienced in this, but if renting a kid's wagon is an option -- I'd think that one with somewhat larger wheels than a standard issue wagon would be a bit easier to use - especially through the inevitable soft or damp salt. Lawn chair, cooler, shade!! -- all are important, the shade more than others. If you'll use the tarp for shade -- you'll need to rig some supports to hold it in place over your head. Just get an inexpensive "E Z Up" and use it, making sure to take it down each evening. It WILL blow away during the night and maybe during a windy day - unless you've taken it down before leaving the thing.
That's enough for now, but feel free to ask us more questions.
Jon a/k/a Seldom Seen Slim