I can help you with your line. I wrote a book on the Van Slyke family called "The Van Slyke Family in America: A Genealogy of Cornelise Antonissen Van Slyke, 1604-1676 and his Mohawk Wife Ots-Toch, including the story of Jacques Hertel, 1603-1651, Father of Ots-Toch and Interpreter to Samuel de Champlain"
In my 189-page book, I trace Ots-Toch's lineage, and Cornelis' and Ots-Toch's descendants to the mid 1800s. I have included some descendants of Cornelis' nephew Willem Van Slyke aka Willem Neef
Your line is from Willem (the nephew). Since the publication of my first book I have been researching and updating the Van Slyke book. I also commissioned research in the archives in Breuckelen, Netherlands and other areas in the Netherlands, with interesting results!
I have almost finished writing a supplement on the descendants of Cornelis Antonissen Van Slyke with updates, corrections and additions ; and a second complete book on The descendants of Willem Pieterse Van Slyke.
But let me outline your Peter's lineage briefly here for you. I have sources for all my data, but I'll just briefly outline so you can at least see how/where you fit in:
1 Pieter Teunis Van Slyke
...... 2 Willem Pieterse Van Slyke 1641 - 1678 .......... +Baertie - 1699 (This is Cornelis' nephew aka Willem Neef)
Engleltie Van Valkenburg was born in Albany NY in 1695 (I have her bpt from the church records) and she was the d/o of my ancestors Jochem Lambertse Van Valkenburg and Eva Hendrickse Vrooman. There's more on the Vrooman family at http://olivetreegenealogy.com/nn/surnames/vrooman.shtmlhttp://olivetreegenealogy.com/nn/surnames/vrooman.shtml
I can also help you with your Elisabeth Barheit, but I have to dig out my notes, so will get back to you on her lineage. You have a solid Barheit line, as you start with Johanna Hanse Barheit -- but what you may not know is that your Helena/Lena Gardenier was bpt in Kinderhook to Samuel Gardenier and Barentje Barheit. So you have at least 3 Barheit lines, all leading back to the immigrant ancestor.
That's another story too, as I have on the back burner research I did two years ago on the Barheit family of Albany. I'm trying to find time to write up my research as an article or another book. If only there were 40 hours in a day! :-)