Hello Leo!
Here are some quick answers, I hope these will help you:
The official name for this badge in Finnish is Jääkärikunniamerkki. The nearest translation in English is Jaeger badge but Jaeger cross will be also good because it is unofficially called Jääkäriristi (=Jaeger cross). But as I mentioned, the Jaeger badge is the most "official" in English.
The Jaeger badge is Finnish badge. It was designed by Mrs. Ellinor Ivalo and its first issued was on February 24th, 1919.
The badge was given for everyone who was served in battallion, both Finnish and Germans ( at least there were some whom were not awarded the badge because a bad behavior, for example in political opinions etc.).
The badge was awarded from "the women of Finland" and it was accepted as an official award. About 1800 badges awarded.
The badges were numbered (later some replacement may have been issued without but not many) and the attachment was with two needles. Later that was changed for screwback in many badges.
The Jaeger badge is quite often availeable in auctions, so it not very hard to get. The price has been somewhere around 1000-1500 euros depending the contion of the badge and to who it was awarded.
Regards
Jussi-Pekka