The "proper" way to do this would, I suppose, be to turn on the numbering of the equations you want to number. You can do this by right clicking the equation and selecting Number this formula. There is also a keyboard shortcut, Alt + M N (hold down Alt, press M, release both, press N), which will work if the cursor is inside in equation.
Of course this is tedious work if you have a lot of equations, so it is better to number as you go. Either by adding a normal displayed equation and numbering as mentioned above, or inserting a numbered equation directly, with Insert --> Math --> Numbered formula.
This is also quite tedious, Ctrl+Shift+M is much easier. What one can do is to modify this shortcut to insert a numbered formula instead, or add a new shortcut for this purpose. Changing keyboard shortcuts are done in Tools --> Preferences --> Editing --> Shortcuts, see e.g. Shortcuts for Lemma, Claim, Theorem etc for details. A function you can use for such a shortcut is
command-sequence math-mode on; math-mutate equation;math-number-toggle
Again though, this doesn't help if one already has written the entire document with unnumbered equations. Then the trick mentioned in the LyX wiki, that you seem to have found, is more convenient: go to Document --> Settings --> LyX preamble and add the following two lines:
\renewcommand\[{\begin{equation}}
\renewcommand\]{\end{equation}}
This won't add numbering of the equations within LyX, but in the PDF they will be numbered.
It depends on what you want. If there is a lot of text between the two points, and the enumerate is somehow "within the text" (if you see what I mean), then you should enumerate-resume as scottkosty says. But if you want the text between the bullet points to be "within the enumerate", then this is supported in LyX as follows, without the need for extra modules or tricks.
Do as you did before:
1. Number 1.
Paragraph after number 1.
1. Number 2.
This should show as above in both LyX and the PDF output. Now put your cursor into the paragraph after number 1 and press tab. Now you will see this:
1. Number 1.
Paragraph after number 1.
2. Number 2.
It will look like this in both LyX and the PDF output.
Best Answer
This is a long-standing bug in LyX. The reason it exists is that when you add an equation, LyX runs LaTeX in the background to generate a preview of the equation. But because it does not want to rerun the previews of all of the other equations (because of performance considerations), only the new equation is compiled and thus there is one equation and it receives a "1". There are a few workarounds: