When I try to split a page in two, right down the middle, I select where I want to split it and no matter where I choose instead of splitting it in two it appears to split it into three parts, where the third portion disappears.

The only way to make it work is to go through a long process that results in double the number of desired pages:

Step 1 – select split page vertically
Step 2 – It does not split it along the line I choose, but instead splits it in 1/3’s

Step 3 NOW – if you then choose to change the page orientation
Step 4 - It somehow reverts it to not looking split, but with two copies of each page

Step 5 - Then if you choose split page vertically
Step 6 - It split it properly, but you now have a each page having two copies.

If you choose to change the orientation before the first split, it doesn’t help at all.

Any ideas out there?

The reason is that the source document has pages with a rotation, and PDFClerk apparently doesn’t handle this case correctly. The solution is to open the source document, export it back out to PDF immediately, and then work with the freshly created copy, which will no longer contain pages with a rotation.