Ok, if I understand you correctly, you've noticed that when you cut/copy and paste something like a fraction then one of the entries gets lost. I can verify this, and it's a bug in LyX; maybe it would be sensible to add it to their bug tracker. However, it only happens in a very specific situation, which you can usually work around. I suspect that this workaround is all you need to know.
If you copy the two entries in something like a \frac by selecting the two individual entries, and then paste those into another \frac, you'll get the behaviour you describe. However, if you select the whole fraction and paste over the top of the whole fraction you're trying to replace, everything works as expected. You can select an entire fraction with either the mouse or the keyboard by starting or ending your selection just outside the fraction rather than going from top to bottom within it; the difference is very visually obvious.
Usually, selecting the whole fraction is easier than selecting the two entries anyway, and has the same result. The one exception is if you want to change the type of the command (e.g. go from \frac to \binom or something). In that case you'll need to cut/copy and paste the two entries individually. A relatively quick way to do this is to copy (or cut) the first entry, and then immediately copy the second entry; then when you move the target location you can paste the second entry as usual and paste the first entry with the "paste recent" function. To do that from the keyboard, press Alt+E (edit menu) then E (paste recent) then down followed by enter.
This would be more annoying for things that accept more than two entries like \matrix and \multline, but doing the same things with those works fine. The caveat here is that there must be already be enough room to paste everything in the clipboard because LyX won't add rows/columns automatically. (If there isn't space then everything is still pasted, but in an odd overlapping way.)