One acre of pen for one cow will be a dry lot in a short amount of time. ( dry lot = bare ground) Electric fence will work fine. As for the preformance in the winter time, KS is pretty well known for having some cold wet winters, and they work well with snow, and with frozen ground. Also, cattle learn quickly with electric fences, and they don't spend a lot of time testing them once they've been bit. I will go for days sometimes not even running my electric fence. They just don't challange them if they have been bit before.
With the comments about the grass clippings, it sounds like you are organic orentated. That's fine, but you will need to feed your cow more than grass clippings, or she will soon become mal nurished. She will require suppliments and a higher protien content supliment. I recomend a molasis lick tub. Also rock salt is important as well.
Cattle don't "crib" like horses do. (if you have ever pryed their mouth open, you will find only one row of teeth in a cow, the bottom, horses have two rows, uppers and lowers. Thats why cattle don't grub pastures down as badly as horses do.) If you decide to put up lumber for a fence, they won't chew it. However they lean with all their bodily weight against something if they can maybe reach a little bit of grass on the other side. (with the exception of new barbed wire, and electric fences) I would be concerned that a 2X4 wouldn't be substantial enough. 2X6 maybe. Also make sure that when you place the boards for cattle you place them on the inside of the posts, so when they are pushed against, they push on the posts, and they don't just pull the nails out.
You also mentioned not wanting to spend a lot of money on the containment. People here mention buying treated posts for fence, sounds expensive. My area, lot of Hedge (osage orange) trees are used for posts. They will last 50 years easliy in the ground without being treated. Also I have seen Railroad ties used for corrall posts. They are heavy, pretreated with creasoat, and will last 20 years without too much problem. You might be able to find them for free, or cheep.