Not sure that I really buy them landing either - in my subjective experience, most of the difficult to explain sightings are aerial, by pilots and radar which have tracked *something* for a while.If the are FTL travelers, then there might be some reason for them to come here (ie, they can't intercept our communications and beam them faster than light but they can travel faster than light). Even so, I can't imagine why they would need to land anywhere. Come to the solar system, listen to our communications, leave. Easy cheesy. Landing in a field in Arkansas seems a little unnecessary for your speculated purpose.
I can see some logic to occasional landings in secluded areas where they don't think anyone will notice: collection of local biological material. They may want grass clippings, bits of tree bark or leaves, needles, some insects and small animals etc. They can disassemble such things, establish what kinds of proteins life on this planet uses. Sequence the DNA and see if all the species here are related to each other, and if they have any genetic resemblance to species elsewhere (ie. was there any "seeding"). See what kind of metabolic processes they use, and if they could live on other planets. Find out what kind of microbes are common on the planet, and whether they pose the risk of a terrible plague to you, or if you can relatively easily design antibiotics/antibodies that neutralize them. Even if they're primarily interested in us as a civilization, you can learn a lot about a species by studying its environment. The environment it evolved in may predispose it to aggression or passivity, or various other things.
Making crop circles in fields seems an odd way to go about this, plus I think various methods to create hoax crop circles have been demonstrated, so I don't really think crop circles are examples of this. However, when you have reasonably credible witnesses who claim to have stumbled onto something sitting on a secluded road or path at night which had bright lights and then shot up into the air... well, IF it was an alien probe or ship, the above is what it MIGHT have been doing.