Greetings Jerry -
I have a Katerina Lapice in my tree also. The information I have is only about her parents and siblings.
This Katerina was born 3 November 1828 in Jednoty house number 57 in Novosedly parish to Jan Lapice, son of Antonin Lapice and Marie, daughter of Vojtech Fiser. Her mother was Mariana, daughter of Mariana Lesanovska.
Katerina would have been the first cousin to my ancestor Jan (John) Lapic(e), his grandfather was Antonin Lapice mentioned above. My John Lapic was one of the first immigrants to Wheatland township and Veseli village in 1863.
I can only assume that my John Lapic and your Katherine Lapic Pavek are related to each other. Lapic is an uncommon name and for both Katherine and John to be in Rice County at the same time is to much of a coincidence.
Most of my ancestors came from the Scott County and Rice County area. My gradmother was a Lapic, her family moved to Lankin, North Dakota in the early 1900's.
I will be more than happy to share other names and information I have on the Lapic family. You can email me privately if you wish.