I use OpenOffice daily. And the answer to your question is "Yes". OpenOffice reads and writes Microsoft format files just fine.
But, let me clarify, the term "deleted from WordPad and MS Works" isn't correct.
A file is a file on its own (it isn't 'inside' one program or another. A file has a format, but using the file in one program doesn't prevent it from being used in another, and a file will not be deleted unless you specifically request it be deleted. Of course, if they are important files, you should have them backed up anyway... but once you do this (i.e. make copies of the files that you move somewhere else) then there is absolutely no risk to using any new Word Processor.
Now the details.
Microsoft writes files that are called something like file1.doc. These files are in Microsoft format.
OpenOffice can read files Microsoft format just fine (I haven't had a problem with this in years). OpenOffice can also write files in Microsoft format. I do this professionally... I write files in OpenOffice and save them in Microsoft format for my Windows-addicted friends. I have never had a problem (and people don't even know I don't use Microsoft).
OpenOffice has a different format as well called ODT. (Actually this is an open document format that was agreed upon by everyone in the world except Microsoft, but Microsoft is big enough to thumb its nose at the rest of thw World).
Microsoft programs don't read this other format (out of sheer stubbornness)-- so if you want to share files with the non-computer saavy, you will want to save your files in DOC format. This is not a problem.
It is pretty easy to make OpenOffice read and write DOC files by default.
So short answer, back up your files, then try OpenOffice. There is no risk and many of us really like it.