(1) Yes. Edit -> Find&Replace(Advanced). You can write math formulas as if you are entering into LyZ math.
(2) Yes. This is called version control. Two ways that I know of are:
-you can use LyX internal version control, see here (this one I did not use before)
-you can use external ones (e.g. Git, I have used it and am loving it.)
(3) Version control can also do this.
(4) Sometimes when an opened file that you are viewing or editing is modified by another program, while you are working on it. The viewer/editor of that file will tell you that the file is modified, and it may ask you if you want to view the new version. As far as I know, Notepad++, Textpad, TeXMaker, Matlab editor are examples of this type of behavior. At this point, if you want to save the old version, you should select the no option and save the file with a different name. If you do not mind loosing the older version, you can select the yes option. Then, the older version will be lost and you will see the modified version.
Sometimes, it just gives you the warning, but does not allow you save the older version.
Sometimes, it does not give a warning at al. This is really what you want if you are previewing the PDF file of your LaTeX document.
In LyX, this warning may be associated with the PDF viewer. In LyX, If you select preview option to view the PDF file of your document, and if the PDF viewer is already open and showing the older version of PDF at that specific moment, this may happen. However, it is not really practical to see this message whenever you compile your LyX file.
Some PDF viewers will not allow the documents to be modified (e.g. in Windows, PDF-XChange) at all, and are very impractical to be used with Lyx. Not suggested.
I prefer SumatraPDF, which does not give this warning and updates the PDF right away. This is really practical and fast. There may be other programs that have this behavior.
This message may very well be related to LyX itself. If same LyX file is opened in two separate LyX programs, and if you modify the file in one of the window, the other window may give you this type of warning:
If you select the cancel option, you may have a chance to save two different versions with different names.
To avoid this issue and possibly loosing data, one is not required to use a version control system. The author just needs to be careful with the file he is working on and be organized.
Version control systems, help to take the snapshots of your document as you develop it. If you did not take the snapshot of your file at the right time, you may still have the above warning and loose information.
On the other hand, I found a version control system such as Git very useful and easy to operate, although it requires some reading at the beginning.