Emulator

Can you please test it again in current development version in OGL or DX11 backend?

Confirm this issue, was presented before and still are now on new builds on all three backends

The per-pixel depth option was removed (so it's always enabled) in 3.5-146 (rb06f30f8452a); as of 3.5-549 (r642eab92bc8c), depth is only calculated per-pixel if depth textures are used (because apparently PPD makes Nvidia cards unable to do a certain optimization or two, causing quite a slowdown). If it works fine with PPD enabled but doesn't in revisions past 3.5-549, either the method of detection for depth textures is failing or the game's using something other than depth textures that makes it require per-pixel depth. (I'm not a graphics dev, please disregard any idiocy.)

Also, OP, I think you may have mixed up the 3.0-xxx and 3.5-xxx revisions. Make sure you use the latest from here: http://dolphin-emu.org/download/

In newest version:
menus act as if per-pixel depth option is always enabled (as pointed out by pauldacheez) however the issue is now compounded. Textboxes no longer show up at all. The effect now covers the entire screen making the menus pretty much unusable.
http://img825.imageshack.us/img825/8923/okamimenu.png

yes, this does occur in all backends

All my revisions are from http://dolphin-emu.org/download/

So, this option is created again and called "Fast Depth Calculation" in 3.5-1344.

Can you please recheck it both with this option enabled/disabled?

Confirmed fixed.

Weird though:
Problem is gone regardless of "Fast Depth Calculation" setting. Both enabled and disabled dont have the problem anymore.

Fast Depth Calculation uses a slightly different method than Disable Per-Pixel Depth did. Evidently it's faster and less problematic at the same time.