Will Pages 09, version 4.3 be lost when upgrading to latest OS X? And, can old pages files be opened in new version of pages?

I have Pages '09 v4.3 on OS X 10.11.5. Before El Capitan, this was a Mavericks machine, and I did not reinstall Pages '09 v4.3. However, you are talking about migrating Pages '09 documents, and the application itself to the new MacBook Air. That is a different story.

If your current Pages '09 v4.3 was installed on your 2014 MacBook Air from the retail iWork '09 DVD, then it will need to be re-installed on the new MacBook Air from that DVD, and then updated with the Apple iWork 9.3 updater, for compatibility with El Capitan. This is because the iWork '09 DVD source of Pages '09, Numbers '09, and Keynote '09 installs dependent libraries in your System areas, and these cannot be reinstalled from a Time Machine or other backup.

If your Pages '09 v4.3 was originally purchased via the OS X App Store, and later updated from same, then it is portable, does not install dependent libraries, and can be copied from your older MacBook Air to the newer one.

In either case, you want to collect your custom templates from your home directory : Library : Application Support : iWork folder, and put it in the same location on your new MacBook Air. Additionally, copy following preference list files and put on a USB stick to transfer to their same location on your new MacBook Air.

1. home directory : Library : Preferences : com.apple.iWork.Pages.plist
2. home directory : Library : Preferences : com.apple.iWork.Numbers.plist
3. home directory : Library : Preferences : com.apple.iWork.Keynote.plist

You want these because they contain all of your iWork '09 application settings, including your custom Symbol and Text Substitution characters from the respective application Preferences panels. As you know, starting with OS X Mavericks, this functionality is now set in the System Preferences : Keyboard : Text panel, and without the above files, your custom settings for the iWork '09 applications are lost.

Although Pages v5.6.2 (current) on El Capitan can generically open Pages '06, '08, and '09 documents, it continues to lack scores of features that are in Pages '09 v4.3, and those missing features will be selectively removed/mangled by Pages v5.6.2. The same caveats apply as they did on Mavericks. Other than the older document open support, nothing concrete has been done to improve Pages v5.6.2 since v5.2 on Mavericks — so Pages '09 v4.3 is still the more usable for realistic processing tasks. Documents created by Pages v5.6.2 are not directly backwards compatible with Pages v5.2.2 on Mavericks, either. My rule of thumb is never open an original Pages '09 document that you wish to keep intact, and compatible, with any release of Pages v5.

It is time that you strongly consider future proofing your Pages '09 (or older) document collection. Export them to Word (content) and PDF (layout) for preservation. Pages '09 v4.3 is living on borrowed time as it was discontinued by Apple in 2013. Apple is not taking any precautions to avoid bricking Pages '09 v4.3 in a future OS X update. If that were to occur, you could only open documents in Pages v5, at no small formatting risk. With Word docs you have a migration opportunity to alternative word processing solutions, including Office 2016 for Mac.

There are separate Pages, Numbers, and Keynote communities in the Apple Support Communities for any of your future questions.

I have requested that your post be moved over into the Pages for Mac Support Community.

The simple solution is to buy a real Word Processor, many good ones are free and provide a level of stability unheard of by Pages

You can recover the Pages '09 documents from Time Machine, but would also advise keeping final versions on a separate, non-bootable, non-cloud based media too. Since you remain on the same machine, the only use for a USB stick is secondary Pages '09 document backup.