In America, this is a Southern colloquial term. It means "to have an opinion on something." It can be used more formally but I haven't heard it that way. Someone can say, "I reckon the sun will set soon" instead of "I think the sun will set soon." They're interchangeable.

Yes, "reckon" basically means calculate, or come to a decision through calculation. It's often used in Australian English to mean "think" (ie. as in "have an opinion about something"). Either way, it still suggests some mental process instead of a random idea.

In Australian English, we always use this word in spoken language. Example: I reckon he she likes you. I reckon, you should tell him the truth.