Not sure about where to get the latest version. Therefore, yes, you need to install it if you want to fix these errors. The errors can only be fixed by editing the ESM/ESP files, or by supplying a mod that overwrites the erroneous data with correct data. These are errors in the game files themselves, which OpenMW will faithfully read and recreate, as it's supposed to. These are not problems with the engine itself, and as such they are not fixed by OpenMW.
These are things like poorly written scripts, misplaced objects, incorrect item/spell stats, et cetera. However, in addition to bugs in the original game engine, Morrowind also has bugs in the game files. Any errors in the programming of the Morrowind engine are a nonissue here, and as such something like the Morrowind Code Patcher is unnecessary. While not quite feature complete (it's very close, though, and fully playable in its current state), it does fix a lot of bugs. OpenMW is a reimplementation of the Morrowind engine.
