In the US, you have to make do with what you can get... and "farmer's cheese" differs a lot throughout the US. In many places, it's pretty close to quark (when crumbled and with a little hydration added, at least).
The only big difference in preparation in many parts of the US (especially those with Eastern or Central European heritage) is how long it's pressed. It's otherwise the same thing: acid-curdled dairy with most of the moisture removed. A little Wondra or 00 flour helps if it won't take the right consistency.
As for the pit, it'd be fine either way. It's a bit easier to form with the pit in, and won't hurt the flavor at all, but removing it would be just fine too.
Yep, in Polish it's "twaróg" and as far as I can tell it doesn't exist in the Anglosphere, the closest word to it is "quark" but for instance British quark has absolutely nothing to do with Twaróg. In Polish it's also called "Biały ser"(white cheese) even though it has nothing to do with cheese.
And yes, you'd need to be some kind of a monster to leave the pit in the fruit used for dumplings.
1. What you call "farmer's cheese" is "tvaroh" which is called "quark" in English. It's not cheese, that would be gross.
2. Leaving the pit inside of the fruit that you wrap in a dumpling is sacrilegious. The fruit should be pitted!