All versions are correct but there could be a difference between 1 and 2. 2 says there's a single forest of this area. 1 doesn't say if it is a single forest or several forests. Is and are are both technically correct as we can think of the square km as being countable or as a continuous (uncountable) area. There isn't really any distinction between the two, though, and most natives would use 'is' as we usually think of area as uncountable. Compare it to money: There is/are $12 in the account. - It would sound very strange to use 'are' here. If you want to say that there is more than one forest, you could put: There is a total of six square kilometers of peach forest in the village. - This would imply that there's more than one area of forest.

‘Forest’ seems wrong . Peaches are usually grown in ‘orchards’. Of course we always say ‘there’s’ or ‘there’re’. You can always say ‘there’s’ regardless of whether the following is singular or plural. Maybe not in an exam, but in real life.

#1 is correct if you use the word "are." The plural form is used because the actual word it is referring to is "kilometers." #2 is also correct, although in my personal opinion it would be better to say "There is a peach forest, over six kilometers in size, in the village." There is a small difference in meaning. If the village contained three separate tracts of peach forest, with a total area of over six square kilometers, then #1 would be correct, but #2 would not be correct. "Forest" can be a countable noun for an unbroken wooded area, or an uncountable noun for "wooded land." Usually, small integer numbers are usually written out in words rather than numerals. Authorities differ on "how small is small." Some say to spell out from one to ten, some say one to twenty, some say one to a hundred. Finally, another unit of area is the "hectare." A square kilometer is 100 hectares. I don't know the rules or customs regarding which to use in a case like yours.

I am not native, but I would use the first sentence, using "are" instead of "is". On the other hand, are you sure "a forest of peaches" is correct? I understand a forest is an extense group of trees that grow spontaneously, but peaches are usually grown by people. I am not sure of the name you should use, perhaps a peach crop? Please check it.

М-да!